Free Math Worksheets

- Identify how the two marked angles are positioned relative to the transversal. - What must be true about this pair of angles when two lines are parallel? - Does the converse of that angle theorem apply?

1. The marked angles, \(70^\circ\) and \(110^\circ\), are same-side interior angles formed by a transversal. 2. Their measures are supplementary because \(70^\circ + 110^\circ = 180^\circ\). 3. By the converse of the same-side interior angles theorem, \(g \parallel h\).

Yes. The \(70^\circ\) and \(110^\circ\) angles are same-side interior angles, and \(70^\circ + 110^\circ = 180^\circ\). Therefore, \(g \parallel h\).

- What is the measure of the angle that forms a linear pair with the \(120^\circ\) angle? - Which marked or calculated angles form an alternate interior angle pair? - What converse theorem can prove that two lines are parallel?

1. The angle that forms a linear pair with the \(120^\circ\) angle measures \(180^\circ - 120^\circ = 60^\circ\). 2. This \(60^\circ\) angle and the marked \(60^\circ\) angle at line \(h\) are alternate interior angles. 3. Since the alternate interior angles are congruent, the converse of the alternate interior angles theorem gives \(g \parallel h\).

Yes. The alternate interior angles both measure \(60^\circ\), so \(g \parallel h\).

- What is the sum of the measures of a linear pair? - Which marked angles lie outside the parallel lines on opposite sides of the transversal? - What is true about alternate exterior angles when lines are parallel?

1. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(m\angle \beta = 180^\circ - 132^\circ = 48^\circ\). 2. Since \(a \parallel b\), \(\alpha\) and \(\delta\) are alternate exterior angles. Therefore, \(m\angle \delta = 132^\circ\).

\(m\angle \beta = 48^\circ\) \(m\angle \delta = 132^\circ\)

- How are the two marked angles positioned relative to the transversal? - What converse theorem proves that two lines are parallel? - Are the marked angle measures equal?

1. Angles \(\alpha\) and \(\beta\) are corresponding angles formed by a transversal. 2. Both angles measure \(74^\circ\). 3. By the converse of the corresponding angles theorem, \(g \parallel h\).

Yes. Since the corresponding angles satisfy \(\alpha = \beta = 74^\circ\), \(g \parallel h\).

- What angle relationship do \(\alpha\) and \(\beta\) have? - What must be true about corresponding angles when two lines are parallel? - Compare the two measures.

1. Angles \(\alpha = 55^\circ\) and \(\beta = 51^\circ\) are corresponding angles. 2. If \(g\) and \(h\) were parallel, corresponding angles would be congruent. 3. Since \(55^\circ \ne 51^\circ\), the lines are not parallel.

No. The corresponding angles measure \(55^\circ\) and \(51^\circ\), so \(g \not\parallel h\).

- Check each statement using vertical, corresponding, or alternate interior angles. - Look at the triangle below line \(g\). What is the sum of its interior angles? - How is \(\delta\) related to the triangle’s angle at the intersection of the two transversals?

1. Statement a) is true because \(\alpha\) and \(\beta\) are vertical angles. 2. Statement b) is true because \(\alpha\) and \(\epsilon\) are alternate interior angles formed by parallel lines. 3. Statement c) is true because \(\gamma\) and \(\mu\) are corresponding angles formed by parallel lines. 4. Statement d) is true. The interior angle at the top vertex of the lower triangle is vertical to \(\delta\), so it also measures \(\delta\). The triangle angle sum gives \(\alpha + \gamma + \delta = 180^\circ\). 5. From the diagram, \(\alpha = 70^\circ\), \(\gamma = 50^\circ\), and \(\delta = 60^\circ\). Since \(\alpha - \gamma = 20^\circ\), statement e) is false.

e) \(\delta = \alpha - \gamma\) is false.

- If \(g\) and \(h\) were parallel, what would be the measure of the angle alternate interior to the \(35^\circ\) angle? - How is the marked right angle related to the angle at \(S\) inside \(\triangle BSC\)? - Compare the three triangle angles with the triangle angle sum.

1. Assume that \(g \parallel h\). Then the interior angle at \(B\) in \(\triangle BSC\) would be alternate interior to the marked \(35^\circ\) angle, so it would measure \(35^\circ\). 2. The marked right angle at \(S\) is vertical to the angle at \(S\) inside \(\triangle BSC\), so the triangle’s angle at \(S\) measures \(90^\circ\). The angle at \(C\) measures \(58^\circ\). 3. The triangle angle sum would be \(35^\circ + 90^\circ + 58^\circ = 183^\circ\), which is impossible. 4. Therefore, the assumption is false, and \(g\) and \(h\) are not parallel.

No. If \(g \parallel h\), the triangle’s angles would sum to \(35^\circ + 90^\circ + 58^\circ = 183^\circ\), not \(180^\circ\).

- Find the angle that forms a linear pair with the \(135^\circ\) angle. - Use the fact that \(a \parallel b\). - Use \(\beta = 90^\circ\) to relate the two diagonal lines.

1. The angle that forms a linear pair with the marked \(135^\circ\) angle measures \(180^\circ - 135^\circ = 45^\circ\). 2. Since \(a \parallel b\), the first diagonal line also forms a \(45^\circ\) acute angle with line \(b\). 3. Because \(\beta = 90^\circ\), the two diagonal lines are perpendicular. Therefore, the second diagonal line forms a \(45^\circ\) acute angle with \(b\). 4. Angle \(\alpha\) is supplementary to that \(45^\circ\) angle, so \(m\angle \alpha = 180^\circ - 45^\circ = 135^\circ\).

\(m\angle \alpha = 135^\circ\)

- Identify the alternate interior angle pairs formed by \(g\), \(h\), and each transversal. - Which three marked angles are the interior angles of one triangle? - Test each statement with the triangle angle sum.

1. Statement a) is true because \(\alpha_1\) and \(\beta_1\) are alternate interior angles formed by parallel lines. 2. Statement b) is true because \(\alpha_2\) and \(\beta_2\) are alternate interior angles formed by parallel lines. 3. Angles \(\alpha_1\), \(\alpha_2\), and \(\gamma\) are the interior angles of the triangle with its base on \(g\). Therefore, \(\alpha_1 + \alpha_2 + \gamma = 180^\circ\), so statement d) is true. 4. In general, \(\gamma = 180^\circ - (\alpha_1 + \alpha_2)\), not \(\alpha_1 + \alpha_2\). Therefore, statement c) is false.

c) \(\gamma = \alpha_1 + \alpha_2\) is false.

- Identify the corresponding angle pair formed by transversal \(AB\). - What does the converse of the corresponding angles theorem say? - Compare the marked angle measures in each diagram.

1. In a), the interior angle at \(A\) and the exterior angle at \(B\) are corresponding angles formed by transversal \(AB\) with lines \(AD\) and \(BC\). Both measure \(105^\circ\). By the converse of the corresponding angles theorem, \(AD \parallel BC\). Therefore, the quadrilateral could be a parallelogram. 2. In b), the corresponding angles measure \(105^\circ\) and \(100^\circ\). Since they are not congruent, \(AD\) and \(BC\) are not parallel. Therefore, the quadrilateral cannot be a parallelogram.

a) \(AD \parallel BC\), so the quadrilateral could be a parallelogram. b) \(AD \not\parallel BC\), so the quadrilateral cannot be a parallelogram.

- Use the right-angle mark and \(\beta = 56^\circ\) to find the remaining angle along line \(h\). - Compare that angle with \(\alpha\). - Which converse theorem can prove the lines parallel?

1. The two diagonal lines are perpendicular, so the marked angle between them is \(90^\circ\). 2. Along the same side of line \(h\), the three adjacent angles form a straight angle. Therefore, the angle between transversal \(s\) and \(h\) is \(180^\circ - 90^\circ - 56^\circ = 34^\circ\). 3. This angle and \(\alpha = 34^\circ\) are alternate interior angles. By the converse of the alternate interior angles theorem, \(g \parallel h\).

Yes. The alternate interior angles both measure \(34^\circ\), so \(g \parallel h\).