Free Math Worksheets

- Find the missing acute angle in each right triangle. - What length scale factor corresponds to multiplying an area by \(4\)? - Every pair of corresponding sides is multiplied by the same scale factor.

1. Each right triangle has a \(90^\circ\) angle. 2. In the first triangle, the third angle is \(180^\circ-90^\circ-32^\circ=58^\circ\). 3. In the second triangle, the third angle is \(180^\circ-90^\circ-58^\circ=32^\circ\). 4. Both triangles have angle measures \(90^\circ,32^\circ,58^\circ\), so they are similar by AA. 5. For part b, the area scale factor is \(k^2=4\), so the positive length scale factor is \(k=2\). 6. The new base length is \(2\cdot6\,\text{cm}=12\,\text{cm}\).

a) The triangles are similar because both have angle measures \(90^\circ,32^\circ,58^\circ\). b) The base of the dilated triangle is \(12\,\text{cm}\).

- Imagine enlarging or reducing a triangle without changing its angles. - What is the difference between similar and congruent figures? - Can two equilateral triangles have different side lengths?

1. Paul’s claim is false. 2. Three matching angle measures determine the shape of a triangle but not its size. Such triangles are similar by AA. 3. For example, two equilateral triangles both have angles measuring \(60^\circ\), \(60^\circ\), and \(60^\circ\). One could have side length \(2\,\text{cm}\), while the other has side length \(5\,\text{cm}\). 4. The triangles have the same shape but different sizes, so they are not congruent. AAA is a similarity condition, not a congruence criterion.

Paul is incorrect. Three pairs of congruent angles guarantee similarity, not congruence. The triangles can have the same shape but different side lengths.

- What is true about the base angles of an isosceles triangle? - What is the sum of the interior angles of a triangle? - What angle condition is sufficient to prove two triangles similar?

1. In isosceles triangle \(ABC\), both base angles measure \(72^\circ\). 2. The vertex angle is \(180^\circ-72^\circ-72^\circ=36^\circ\). Thus, the angle measures are \(36^\circ,72^\circ,72^\circ\). 3. In triangle \(DEF\), the missing angle is \(180^\circ-36^\circ-72^\circ=72^\circ\). 4. Both triangles have the same three angle measures, so they are similar by AA.

Yes. Both triangles have interior angle measures \(36^\circ,72^\circ,72^\circ\), so they are similar by AA.

- What is the sum of the interior angles of a triangle? - Find all three angle measures in each triangle. - Which similarity criterion uses only angle information?

1. In triangle \(ABC\), \(\angle C=90^\circ\) and \(\angle A=40^\circ\), so \(\angle B=180^\circ-90^\circ-40^\circ=50^\circ\). 2. In triangle \(DEF\), one angle is \(90^\circ\) and one acute angle is \(50^\circ\), so the third angle is \(180^\circ-90^\circ-50^\circ=40^\circ\). 3. Both triangles have angle measures \(90^\circ,40^\circ,50^\circ\). Therefore, they are similar by AA.

The triangles are similar by AA because both have angle measures \(90^\circ,40^\circ,50^\circ\).

- What conditions can prove two triangles similar? - What is the third angle in a right triangle with a \(40^\circ\) angle? - For Sophia’s claim, compare the two legs and the angle between them. - Match the shorter leg to the shorter leg and the longer leg to the longer leg.

1. Lucas’s claim is true. A right triangle already has a \(90^\circ\) angle. If another angle is \(40^\circ\), the third angle is \(180^\circ-90^\circ-40^\circ=50^\circ\). Every such triangle has angle measures \(90^\circ,40^\circ,50^\circ\), so the triangles are similar by AA. 2. Sophia’s claim is true. In every such triangle, the included angle between the legs is \(90^\circ\), and the ratio of the two adjacent leg lengths is \(2\) to \(1\). Therefore, any two such triangles are similar by SAS.

Both claims are true. Lucas’s claim follows from AA because every triangle described has angles \(90^\circ,40^\circ,50^\circ\). Sophia’s claim follows from SAS because the included angle is \(90^\circ\) and the adjacent leg-length ratio is always \(2\) to \(1\).

- What is the sum of the interior angles of a triangle? - When do angle measures establish triangle similarity? - What does “isosceles” tell you about the base angles? - Find all three angles before comparing each triangle with the reference triangle.

1. The third angle of the reference triangle is \(180^\circ-48^\circ-72^\circ=60^\circ\). Its angle measures are \(48^\circ,60^\circ,72^\circ\). 2. For part a, the missing angle is \(180^\circ-72^\circ-60^\circ=48^\circ\). The triangle is similar to the reference triangle by AA. 3. For part b, both base angles are \(48^\circ\), so the vertex angle is \(180^\circ-2\cdot48^\circ=84^\circ\). The angle measures \(48^\circ,48^\circ,84^\circ\) do not match the reference triangle, so the triangles are not similar. 4. For part c, the missing angle is \(180^\circ-48^\circ-60^\circ=72^\circ\). The triangle is similar to the reference triangle by AA.

a) Similar b) Not similar c) Similar

- Model each ramp as a right triangle. - Which two sides are given for each triangle? - Compare the rise-to-run ratios. - What similarity criterion uses two proportional sides and the included angle?

1. Both ramp triangles are right triangles, so compare the ratios of the two legs. 2. For Ramp A, the rise-to-run ratio is \(\frac{3}{15}=0.2\). 3. For Ramp B, the rise-to-run ratio is \(\frac{4}{20}=0.2\). 4. The leg ratios are equal and the included angles are both \(90^\circ\), so the triangles are similar by SAS.

Yes. The ramp triangles are similar because both have a rise-to-run ratio of \(0.2\).

- Find the third angle in each triangle. - Think about the side lengths after an equilateral triangle is cut in half. - What happens to a fraction when its numerator and denominator are multiplied by the same number?

1. Each triangle has angles of \(90^\circ\), \(30^\circ\), and \(60^\circ\), so the triangles are similar by AA. 2. Let the equilateral triangle have side length \(c\). Cutting it in half creates a right triangle whose hypotenuse is \(c\) and whose side opposite the \(30^\circ\) angle is \(\frac{c}{2}\). 3. Thus, \(\frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{c/2}{c}=\frac{1}{2}\). 4. Doubling all side lengths multiplies both parts of the ratio by \(2\), so \(\frac{2a}{2c}=\frac{a}{c}\).

a) The triangles are similar by AA because both have angles of \(90^\circ\), \(30^\circ\), and \(60^\circ\). b) The ratio is \(\frac{1}{2}\). c) Scaling multiplies both the numerator and denominator by the same factor, so the ratio does not change.

- Which angle is shared by the two triangles? - What angle is formed where the altitude meets the hypotenuse? - What can you conclude when two pairs of corresponding angles are congruent?

1. In the original triangle, \(\angle C=90^\circ\). 2. Triangle \(ADC\) shares \(\angle A\) with triangle \(ABC\). 3. Since \(CD\perp AB\), \(\angle ADC=90^\circ\). 4. The two triangles have two pairs of congruent angles, so \(\triangle ADC\sim\triangle ACB\) by AA.

Triangle \(ADC\) is similar to triangle \(ABC\) by AA because they share \(\angle A\) and each has a right angle.

- What is the sum of the interior angles of a triangle? - Which angle condition proves triangles similar? - What properties are preserved when a figure is enlarged by a dilation?

1. The missing angle in Triangle 1 is \(180^\circ-42^\circ-75^\circ=63^\circ\). 2. The missing angle in Triangle 2 is \(180^\circ-75^\circ-63^\circ=42^\circ\). 3. Both triangles have angle measures \(42^\circ,63^\circ,75^\circ\), so they are similar by AA. 4. Doubling every side length is a dilation with scale factor \(2\). A dilation preserves angle measures, so the new triangle has the same interior angles as the original.

a) Yes. Both triangles have angle measures \(42^\circ,63^\circ,75^\circ\), so they are similar by AA. b) All interior angle measures remain unchanged because a dilation preserves shape and angle measure.

- Match the shortest, middle, and longest sides. - What must be true about all three corresponding side-length ratios? - Calculate one ratio for each pair of corresponding sides.

1. Match the sides from shortest to longest: \(c=6\,\text{cm}\) with \(f=4\,\text{cm}\), \(a=9\,\text{cm}\) with \(d=6\,\text{cm}\), and \(b=12\,\text{cm}\) with \(e=8\,\text{cm}\). 2. The ratios are \(\frac{c}{f}=\frac{6}{4}=1.5\), \(\frac{a}{d}=\frac{9}{6}=1.5\), and \(\frac{b}{e}=\frac{12}{8}=1.5\). 3. All three corresponding side-length ratios are equal, so the triangles are similar by SSS.

Yes. The triangles are similar by SSS because every corresponding side-length ratio equals \(1.5\).

- What is the sum of the interior angles of a triangle? - Find the missing angle in each triangle. - When can angle measures prove two triangles similar?

1. In triangle \(PQR\), \(\angle R=180^\circ-52^\circ-68^\circ=60^\circ\). 2. In triangle \(STU\), \(\angle S=180^\circ-68^\circ-60^\circ=52^\circ\). 3. Both triangles have angle measures \(52^\circ,60^\circ,68^\circ\), so they are similar by AA.

Yes. The triangles are similar by AA; \(\angle R=60^\circ\) and \(\angle S=52^\circ\).

- Which pair of angles is marked congruent? - What is true about opposite angles formed by intersecting lines? - Which triangle similarity criterion uses two angle pairs?

1. It is given that \(\angle ABE\cong\angle CDE\). 2. Angles \(\angle AEB\) and \(\angle CED\) are vertical angles, so they are congruent. 3. Therefore, \(\triangle ABE\sim\triangle CDE\) by AA.

The triangles are similar by AA because \(\angle ABE\cong\angle CDE\) and \(\angle AEB\cong\angle CED\) as vertical angles.

- Which angles are known from the fact that both triangles are right triangles? - How many congruent angle pairs are needed for AA similarity? - Use the given angle congruence.

1. Since the triangles are right triangles, \(\angle ACE\cong\angle EKF\), and both measure \(90^\circ\). 2. It is given that \(\angle CAE\cong\angle FEK\). 3. Therefore, \(\triangle ACE\sim\triangle EKF\) by AA.

The triangles are similar by AA because they have congruent right angles and \(\angle CAE\cong\angle FEK\).

- Which angle belongs to both the smaller and larger triangle? - Which angle congruence is given? - Which similarity criterion uses two angle pairs?

1. Compare triangles \(BKP\) and \(BAC\). 2. The angle at \(B\) is shared: \(\angle KBP\cong\angle ABC\). 3. It is given that \(\angle BKP\cong\angle BAC\). 4. Therefore, \(\triangle BKP\sim\triangle BAC\) by AA.

\(\triangle BKP\sim\triangle BAC\) by AA.

- Which two side lengths are adjacent to the given angle in each triangle? - Compare the ratios of the corresponding side pairs. - Which similarity criterion uses two proportional side pairs and the included angle?

1. Compare the pairs of sides adjacent to the given angles: \(\frac{AB}{DE}=\frac{12}{8}=1.5\) and \(\frac{BC}{EF}=\frac{15}{10}=1.5\). 2. The two pairs of corresponding sides are proportional. 3. The included angles are congruent because \(\angle B=\angle E=38^\circ\). 4. Therefore, \(\triangle ABC\sim\triangle DEF\) by SAS similarity.

Yes. \(\triangle ABC\sim\triangle DEF\) by SAS similarity because \(\frac{AB}{DE}=\frac{BC}{EF}=1.5\) and the included angles \(\angle B\) and \(\angle E\) are congruent.

- What is the sum of the angle measures in a triangle? - How are the base angles of an isosceles triangle related? - Which similarity criterion uses two pairs of congruent angles?

1. In an isosceles triangle, the two base angles are congruent. 2. In each triangle, each base angle measures \(\frac{180^\circ-44^\circ}{2}=68^\circ\). 3. Both triangles therefore have angle measures \(44^\circ\), \(68^\circ\), and \(68^\circ\), so they are similar by AA.

Yes. Both triangles have angle measures \(44^\circ\), \(68^\circ\), and \(68^\circ\), so they are similar by AA.

- Which angle is shared because \(D\) lies on \(AC\)? - Which second pair of congruent angles is given?

1. Because \(D\) lies on \(AC\), \(\angle BAD\cong\angle CAB\). 2. It is given that \(\angle ABD\cong\angle ACB\). 3. Therefore, \(\triangle ABD\sim\triangle ACB\) by AA.

\(\triangle ABD\sim\triangle ACB\) by AA.

- What do the matching tick marks tell you about each triangle? - Find the two acute angle measures in an isosceles right triangle. - Which similarity criterion uses angle measures?

1. Each triangle is a right triangle with two congruent legs, as shown by the tick marks. 2. In each isosceles right triangle, the two acute angles are congruent and have measure \(\frac{180^\circ-90^\circ}{2}=45^\circ\). 3. Both triangles have angle measures \(45^\circ\), \(45^\circ\), and \(90^\circ\), so they are similar by AA.

Yes. The triangles are similar by AA because both are \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangles.

- Match the shortest sides, the middle-length sides, and the longest sides. - Check whether one constant scale factor relates all three pairs.

1. Match the sides from shortest to shortest, middle to middle, and longest to longest. 2. The corresponding ratios are \(\frac{6}{4}=1.5\), \(\frac{9}{6}=1.5\), and \(\frac{11}{8}=1.375\). 3. The ratios are not all equal, so the triangles are not similar.

No. The triangles are not similar because the corresponding side-length ratios are \(1.5\), \(1.5\), and \(1.375\), which are not all equal.

- What angle is formed by a segment perpendicular to a line? - What does an angle bisector do to an angle? - Which similarity criterion uses two pairs of congruent angles?

1. Because \(BB_1\) and \(CC_1\) are perpendicular to the angle bisector, \(\angle AB_1B\) and \(\angle AC_1C\) are right angles. 2. The angle bisector divides \(\angle A\) into two congruent angles, so \(\angle BAB_1\cong\angle CAC_1\). 3. Therefore, \(\triangle ABB_1\sim\triangle ACC_1\) by AA.

\(\triangle ABB_1\sim\triangle ACC_1\) by AA because each triangle has a right angle and the acute angles at \(A\) are congruent.

- Compare the ratio of the two legs in each right triangle. - What angle is included between the two legs? - How can the two hypotenuse lengths be used to find the scale factor?

1. In the first triangle, the ratio of the legs is \(\frac{1}{2}\). 2. In the second triangle, the ratio of the legs is \(\frac{5}{10}=\frac{1}{2}\). 3. The included angle between the legs is \(90^\circ\) in both triangles. Therefore, the triangles are similar by SAS similarity. 4. The hypotenuse of the second triangle is \(c_2=\sqrt{5^2+10^2}=\sqrt{125}=5\sqrt{5}\,\text{cm}\). 5. The scale factor from the first triangle to the second is \(k=\frac{c_2}{c_1}=\frac{5\sqrt{5}}{2\sqrt{5}}=\frac{5}{2}=2.5\).

a) Yes. The triangles are similar because both are right triangles and their corresponding legs are proportional. b) The scale factor from the first triangle to the second is \(k=2.5\).

- Find the vertex angle of the first triangle. - Use the vertex angle of the second triangle to find its two base angles. - How is the area scale factor related to the length scale factor for similar figures?

1. In the first triangle, both base angles are \(70^\circ\), so the vertex angle is \(180^\circ-70^\circ-70^\circ=40^\circ\). 2. In the second triangle, the two base angles share the remaining \(180^\circ-40^\circ=140^\circ\), so each base angle is \(\frac{140^\circ}{2}=70^\circ\). 3. Both triangles have angle measures \(70^\circ,70^\circ,40^\circ\), so they are similar by AA. 4. The area scale factor is \(k^2=2.25\), so the length scale factor is \(k=\sqrt{2.25}=1.5\). 5. The base of the larger triangle is \(1.5\cdot5\,\text{cm}=7.5\,\text{cm}\).

a) Yes. Both triangles have angle measures \(70^\circ,70^\circ,40^\circ\). b) The base of the larger triangle is \(7.5\,\text{cm}\).

- Find the missing interior angle in each triangle. - The area ratio of similar figures is the square of the length scale factor. - Perimeter changes by the same factor as corresponding side lengths.

1. The missing angle of the first triangle is \(180^\circ-40^\circ-80^\circ=60^\circ\). 2. The missing angle of the second triangle is \(180^\circ-80^\circ-60^\circ=40^\circ\). 3. Both triangles have angle measures \(40^\circ,60^\circ,80^\circ\), so they are similar by AA. 4. For part b, \(\frac{A_{\text{large}}}{A_{\text{small}}}=\frac{49}{25}=k^2\). 5. The positive scale factor is \(k=\sqrt{\frac{49}{25}}=\frac{7}{5}=1.4\). 6. Perimeter scales by the length scale factor, so \(P_{\text{large}}=1.4\cdot20\,\text{cm}=28\,\text{cm}\).

a) Yes. Both triangles have angle measures \(40^\circ,60^\circ,80^\circ\). b) The scale factor is \(k=\frac{7}{5}=1.4\), and the perimeter of the larger triangle is \(28\,\text{cm}\).

- Use the \(180^\circ\) triangle angle sum. - What does matching angle measures tell you about similarity? - Does angle information alone determine a triangle’s size?

1. For Triangle 1, \(\gamma_1 = 180^\circ - (45^\circ + 75^\circ) = 60^\circ\). 2. For Triangle 2, \(\beta_2 = 180^\circ - (45^\circ + 60^\circ) = 75^\circ\). 3. Both triangles have angle measures \(45^\circ\), \(60^\circ\), and \(75^\circ\), so they have the same shape and are similar by AA. 4. No side length is given, so the triangles may have different scale factors. Therefore, they are not necessarily congruent.

a) \(\gamma_1 = 60^\circ\) and \(\beta_2 = 75^\circ\) b) The triangles have the same angle measures, so they have the same shape and are similar. c) No. They may be different sizes because no corresponding side lengths are specified.

- Which missing side lengths are needed before all ratios can be compared? - Use the Pythagorean theorem in each right triangle. - How can side lengths show that one triangle is a scaled copy of another?

1. In triangle \(A\), the hypotenuse is \(c_1=\sqrt{6^2+8^2}=\sqrt{100}=10\,\text{cm}\). 2. In triangle \(B\), the missing leg is \(b_2=\sqrt{20^2-12^2}=\sqrt{256}=16\,\text{cm}\). 3. Compare corresponding sides: \(\frac{a_2}{a_1}=\frac{12}{6}=2\), \(\frac{b_2}{b_1}=\frac{16}{8}=2\), and \(\frac{c_2}{c_1}=\frac{20}{10}=2\). 4. All three corresponding side-length ratios are equal, so the triangles are similar by SSS similarity.

Yes. Triangle \(A\) has side lengths \(6\,\text{cm}\), \(8\,\text{cm}\), and \(10\,\text{cm}\), and triangle \(B\) has side lengths \(12\,\text{cm}\), \(16\,\text{cm}\), and \(20\,\text{cm}\). Every side of triangle \(B\) is twice the corresponding side of triangle \(A\).

- Use \(A=\frac{1}{2}ab\) to choose two different pairs of positive leg lengths with the required area. - What must be true about corresponding side-length ratios for the triangles to be similar? - Can right triangles with equal areas have different shapes?

1. For a right triangle with leg lengths \(a\) and \(b\), the area is \(A=\frac{1}{2}ab\). 2. A right triangle with legs \(3\,\text{cm}\) and \(4\,\text{cm}\) has area \(\frac{1}{2}\cdot3\cdot4=6\,\text{cm}^2\). 3. A right triangle with legs \(2\,\text{cm}\) and \(6\,\text{cm}\) also has area \(\frac{1}{2}\cdot2\cdot6=6\,\text{cm}^2\). 4. The leg ratios are \(\frac{3}{4}\) and \(\frac{2}{6}=\frac{1}{3}\), so the triangles are not similar. Equal area does not guarantee similarity.

No. For example, right triangles with legs \(3\,\text{cm}\) and \(4\,\text{cm}\), and with legs \(2\,\text{cm}\) and \(6\,\text{cm}\), both have area \(6\,\text{cm}^2\) but are not similar because their leg-length ratios differ.

- Use the triangle angle-sum theorem. - What angles are created when an altitude is perpendicular to the hypotenuse? - How do the two angles at \(C\) combine in the original right triangle? - Express each unknown angle in terms of \(\alpha\).

1. In triangle \(ADC\), \(\angle ADC=90^\circ\) and \(\angle CAD=\angle A\). If \(\angle A=\alpha\), then \(\angle ACD=90^\circ-\alpha\). 2. In triangle \(CDB\), \(\angle CDB=90^\circ\). Because the original angle at \(C\) is \(90^\circ\), \(\angle DCB=\alpha\), and the remaining angle at \(B\) is \(90^\circ-\alpha\). 3. Both smaller triangles have angle measures \(90^\circ,\alpha,90^\circ-\alpha\), so they are similar by AA. 4. When \(\alpha=35^\circ\), triangle \(ADC\) has angles \(35^\circ,55^\circ,90^\circ\), and triangle \(CDB\) has angles \(55^\circ,35^\circ,90^\circ\).

a) The triangles are similar by AA because each has angle measures \(90^\circ,\alpha,90^\circ-\alpha\). b) Triangle \(ADC\) has angles \(35^\circ,55^\circ,90^\circ\). Triangle \(CDB\) has angles \(55^\circ,35^\circ,90^\circ\).

- What must be true about all corresponding side lengths for two triangles to be similar by SSS? - Find the three side-length ratios. - Does adding the same amount to each side preserve proportionality?

1. The side lengths of \(T_2\) are \(8\,\text{cm}\), \(10\,\text{cm}\), and \(12\,\text{cm}\). 2. Compare corresponding side-length ratios: \(\frac{8}{6}=\frac{4}{3}\), \(\frac{10}{8}=\frac{5}{4}\), and \(\frac{12}{10}=\frac{6}{5}\). 3. The three ratios are not equal, so the corresponding sides are not proportional. Therefore, the triangles are not similar by SSS.

No. The corresponding side-length ratios \(\frac{4}{3}\), \(\frac{5}{4}\), and \(\frac{6}{5}\) are not equal, so the triangles are not similar.

- What conditions define similar polygons? - Is there a triangle similarity criterion based only on angle measures? - Can two quadrilaterals have the same angle measures but different side-length proportions?

1. Similar polygons have congruent corresponding angles and proportional corresponding side lengths. 2. For triangles, the claim is true. If two corresponding angle pairs are congruent, then the triangles are similar by the AA similarity criterion. 3. For trapezoids, the claim is false. Consider two isosceles trapezoids that both have angle measures \(45^\circ,45^\circ,135^\circ,135^\circ\). One can have bases \(6\,\text{cm}\) and \(2\,\text{cm}\) with height \(2\,\text{cm}\), while another has bases \(8\,\text{cm}\) and \(4\,\text{cm}\) with height \(2\,\text{cm}\). Their corresponding base ratios are \(\frac{8}{6}\) and \(\frac{4}{2}\), which are not equal. Therefore, the trapezoids are not similar even though their corresponding angles are congruent.

a) The claim is true for triangles because two pairs of congruent corresponding angles establish similarity by AA. b) The claim is false for trapezoids. Two trapezoids can have congruent corresponding angles without having proportional corresponding side lengths.

- Substitute \(2a\) and \(2b\) into the tangent ratio. - Simplify the new ratio. - What happens to angles when a figure is scaled uniformly?

1. After doubling both legs, \(\tan(\alpha_{\text{new}})=\frac{2a}{2b}=\frac{a}{b}\). 2. The tangent ratio is unchanged, so the acute angle is unchanged: \(\alpha_{\text{new}}=\alpha\). 3. Geometrically, multiplying both legs by the same scale factor produces a triangle similar to the original. Corresponding angles in similar triangles are congruent.

Angle \(\alpha\) does not change. The ratio \(\frac{2a}{2b}\) equals \(\frac{a}{b}\), and the new triangle is a scaled copy of the original.

- What angle relationships hold in an isosceles triangle? - Could the given angle be either a vertex angle or a base angle? - What must be true about corresponding angles for triangles to be similar? - Find the other angles in both possible cases.

1. In an isosceles triangle, the \(50^\circ\) angle could be the vertex angle or a base angle. 2. If \(50^\circ\) is the vertex angle, each base angle is \(\frac{180^\circ-50^\circ}{2}=65^\circ\). The angle measures are \(50^\circ,65^\circ,65^\circ\). 3. If \(50^\circ\) is a base angle, the other base angle is also \(50^\circ\), and the vertex angle is \(180^\circ-2\cdot50^\circ=80^\circ\). The angle measures are \(50^\circ,50^\circ,80^\circ\). 4. Because two different angle sets are possible, the triangles are not guaranteed to be similar.

The \(50^\circ\) angle might be the vertex angle, giving angles \(50^\circ,65^\circ,65^\circ\), or a base angle, giving angles \(50^\circ,50^\circ,80^\circ\). These triangles are not similar, so the given information is insufficient.

- Use the shared angle at \(A\) and the angles created by the parallel segments. - Find the full length \(AB\). - Set up a proportion with corresponding sides.

1. The triangles share angle \(A\). Since \(DE\parallel BC\), a second pair of corresponding angles is congruent. Therefore, \(\triangle ADE\sim\triangle ABC\) by AA. 2. The full side length is \(AB=AD+DB=4+2=6\,\text{cm}\). 3. Corresponding sides satisfy \(\frac{DE}{BC}=\frac{AD}{AB}\). Thus, \(\frac{DE}{7.5}=\frac{4}{6}\). 4. Therefore, \(DE=7.5\cdot\frac{4}{6}=5\,\text{cm}\).

a) \(\triangle ADE\sim\triangle ABC\) by AA. b) \(DE=5\,\text{cm}\)

- Which angles are right angles because \(CD\) is an altitude? - Express the acute angles in terms of one angle in the original right triangle. - Which similarity criterion follows from two pairs of congruent angles?

1. Because \(CD\perp AB\), \(\angle ADC\) and \(\angle CDB\) are both right angles. 2. Let \(\angle CAD=\alpha\). Since \(\triangle ACD\) is a right triangle, \(\angle ACD=90^\circ-\alpha\). 3. In \(\triangle ABC\), \(\angle CBA=90^\circ-\alpha\). Because \(D\) lies on \(AB\), \(\angle CBD=\angle CBA\). 4. Thus, \(\angle ACD\cong\angle CBD\), and the triangles also have congruent right angles. Therefore, \(\triangle ACD\sim\triangle CBD\) by AA.

\(\triangle ACD\sim\triangle CBD\) by AA because both have a right angle and \(\angle ACD\cong\angle CBD\).

- How are an exterior angle and its adjacent interior angle related? - How are the base angles of an isosceles triangle related?

1. In the left triangle, the interior angle adjacent to the \(120^\circ\) exterior angle is \(180^\circ-120^\circ=60^\circ\). 2. Because the left triangle is isosceles, its other base angle is also \(60^\circ\), so its vertex angle is \(60^\circ\). 3. In the right isosceles triangle, the vertex angle is \(60^\circ\). Each base angle is \(\frac{180^\circ-60^\circ}{2}=60^\circ\). 4. Both triangles have three \(60^\circ\) angles, so they are similar by AA.

Yes. Both triangles are equilateral, so they are similar by AA.

- Which angles are congruent because \(AK\) is an angle bisector? - Which two inscribed angles intercept the same arc? - Use two angle pairs to establish similarity.

1. Since \(AK\) bisects \(\angle BAC\) and \(M\) lies on \(AK\), \(\angle BAM\cong\angle KAC\). 2. Because \(M\) lies on \(BC\), \(\angle ABM=\angle ABC\). The angles \(\angle ABC\) and \(\angle AKC\) are inscribed angles that intercept the same arc \(AC\), so they are congruent. 3. Therefore, \(\triangle ABM\sim\triangle AKC\) by AA.

\(\triangle ABM\sim\triangle AKC\) by AA because the angle bisector gives \(\angle BAM\cong\angle KAC\), and the inscribed angle theorem gives \(\angle ABM\cong\angle AKC\).

- Use the triangle angle sum to find both missing angles. - Match corresponding vertices by equal angle measures. - Is the \(6.5\,\text{cm}\) side opposite the same angle in both triangles?

1. In \(\triangle ABC\), \(\beta = 180^\circ - 50^\circ - 70^\circ = 60^\circ\). 2. In \(\triangle DEF\), \(\epsilon = 180^\circ - 60^\circ - 70^\circ = 50^\circ\). 3. Both triangles have angle measures \(50^\circ\), \(60^\circ\), and \(70^\circ\), so they are similar by AA. 4. The side \(b = 6.5\,\text{cm}\) is opposite the \(60^\circ\) angle, while side \(e = 6.5\,\text{cm}\) is opposite the \(50^\circ\) angle. These are not corresponding sides. 5. Because distinct angles must have distinct opposite side lengths, the common value \(6.5\,\text{cm}\) cannot represent corresponding sides in congruent triangles. Therefore, the triangles are not congruent.

The triangles are not congruent. They are similar, but the equal-length sides lie opposite different angle measures and are not corresponding sides.

- Rewrite the product equation as an equation of ratios. - Which angle is shared by the two triangles? - Which similarity criterion combines two proportional side pairs with the included angle?

1. Rewrite the given equation by dividing both sides by \(CB\cdot BK\): \(\frac{AB}{CB}=\frac{BP}{BK}\). 2. The included angle at \(B\) is common to both triangles: \(\angle ABC\cong\angle PBK\). 3. Two pairs of sides that include the congruent angle are proportional, so \(\triangle ABC\sim\triangle PBK\) by SAS.

\(\triangle ABC\sim\triangle PBK\) by SAS because \(\frac{AB}{CB}=\frac{BP}{BK}\) and the triangles share the included angle at \(B\).