Question

In the figure, m<2=92 and m<12=74. Find the measure of each angle. Tell which postulate (s) or theorem (s) you used.
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Answer 1

Answer:

$\angle 10=92\°\\\angle 8 = 92\°\\\angle 9=88\°\\\angle 5=106\°\\\angle 11 = 106\°\\\angle 13 = 106\°$

Step-by-step explanation:

In the figure $m \parallel n$, and r and s are transversal that intercept those parallels.

Now, from the each interception of each transversal we have 8 angles related.

$\angle 1, \angle 2, \angle 7, \angle 8, \angle 9, \angle 10, \angle 16, \angle 15$, all these angles are related.

$\angle 3, \angle 4, \angle 5, \angle 6, \angle 11, \angle 12, \angle 13, \angle 14$, all these angles are related.

By corresponding angles, we have the following congruence

$\angle 2 \cong \angle 10$

This means that

$\angle 10 = \angle 2= 92\°$

$\angle 9 = 180\° - \angle 10$,  by supplementary angles.

$\angle 9= 180\°-92\°=88\°$

$\angle 8 = \angle 2$, by vertical angles theorem.

$\angle 8= 92$

$\angle 5 = \angle 11$, by alternate interior angles.

$\angle 11=180\° - \angle 12$, by supplementary angles.

$\angle 11= 180\°-74\°=106\°$

$\angle 5=106\°$

$\angle 13 = \angle 11$, by vertical angles theorem.

$\angle 13 = 106\°$

Therefore, the answers are

$\angle 10=92\°\\\angle 8 = 92\°\\\angle 9=88\°\\\angle 5=106\°\\\angle 11 = 106\°\\\angle 13 = 106\°$

Answer 2

The measure of the given angles and the theorem used to find each are:

11. $\mathbf{m \angle 10 = 92^{\circ}}$ - corresponding angles theorem

12. $\mathbf{m \angle 8 = 92^{\circ}}$ - vertical angles theorem

13. $\mathbf{m \angle 9 = 106^{\circ}}$ - same-side exterior angles theorem

14. $\mathbf{m \angle 5 = 106^{\circ}}$ - same-side interior angles theorem

15. $\mathbf{m \angle 11 = 106^{\circ}}$ - corresponding angles theorem

16. $\mathbf{m \angle 13 = 106^{\circ}}$ - vertical angles theorem

The relative position that angles formed when a transversal cuts across two lines determines the type of angle pairs two angles are and the theorem that applies for each pair.

Given:

• m<2 = 92
• m<12 = 74

11. m<2 and m<10 are corresponding angles.

Based on the corresponding angles theorem, m<2 = m<10

$\mathbf{m \angle 10 = 92^{\circ}}$

12. m<8 and <2 are vertical angles.

Based on the vertical angles theorem, m<8 = m<2

$\mathbf{m \angle 8 = 92^{\circ}}$

13. m<9 and <12 are same-side exterior angles.

Based on the same-side exterior angles theorem, m<9 = 180 - m<12

$m \angle 9 = 180 - 74\\\\\mathbf{m \angle 9 = 106^{\circ}}$

14. m<5 and <12 are same-side interior angles.

Based on the same-side interior angles theorem, m<5 = 180 - m<12

$m \angle 5 = 180 - 74\\\\\mathbf{m \angle 5 = 106^{\circ}}$

15. m<11 and m<9 are corresponding angles.

Based on the corresponding angles theorem, m<11 = m<9

$\mathbf{m \angle 11 = 106^{\circ}}$

16. m<13 and <11 are vertical angles.

Based on the vertical angles theorem, m<13 = m<11

$\mathbf{m \angle 13 = 106^{\circ}}$

In summary, the measure of the given angles and the theorem used to find each are:

11. $\mathbf{m \angle 10 = 92^{\circ}}$ - corresponding angles theorem

12. $\mathbf{m \angle 8 = 92^{\circ}}$ - vertical angles theorem

13. $\mathbf{m \angle 9 = 106^{\circ}}$ - same-side exterior angles theorem

14. $\mathbf{m \angle 5 = 106^{\circ}}$ - same-side interior angles theorem

15. $\mathbf{m \angle 11 = 106^{\circ}}$ - corresponding angles theorem

16. $\mathbf{m \angle 13 = 106^{\circ}}$ - vertical angles theorem

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