Question

1. x = _ 20) x = _ ii. find the measure of each angle. 21) ∠1 22) ∠2 23) ∠3 24) ∠4 25) ∠5 26) ∠6

Explanation:

Step1: Solve for x in problem 19

Use the property of isosceles - triangles. The two equal - angled triangles in the first figure have equal base - angles. The sum of angles in a triangle is 180°. First, find the third angle of the left - hand isosceles triangle. Since the two base - angles are equal and one of the non - base angles is 56°, the base - angle of the left - hand isosceles triangle is $\frac{180 - 56}{2}=62^{\circ}$. Then, considering the large triangle, the angle adjacent to the 56° angle in the large triangle is 180 - 56=124°. The other two angles of the large triangle are equal. Let the unknown angle be x. So, $x=\frac{180 - 124}{2}=28^{\circ}$.

Step2: Solve for x in problem 20

Use the angle - sum property of a triangle. The sum of angles in a triangle is 180°. In the left - hand triangle, the third angle is $180-(50 + 62)=68^{\circ}$. In the right - hand triangle, the third angle is $180-(53 + 80)=47^{\circ}$. The angle x is found by considering the intersection of the two triangles. Using the property of vertical angles and the angle - sum property of a triangle formed at the intersection, we know that $x = 180-(68 + 47)=65^{\circ}$.

Step3: Find ∠1

∠1 and the 68° angle are vertical angles. Vertical angles are equal. So, ∠1 = 68°.

Step4: Find ∠2

∠2 and ∠1 are supplementary (a straight - line has an angle of 180°). So, ∠2=180 - 68 = 112°.

Step5: Find ∠3

∠3 = 90° (given in the figure).

Step6: Find ∠4

∠4 and ∠3 are supplementary. So, ∠4=180 - 90 = 90°.

Step7: Find ∠5

∠5 and the angle adjacent to the 68° angle in the large triangle are corresponding angles. The angle adjacent to the 68° angle is 112°. So, ∠5 = 112°.

Step8: Find ∠6

∠6 and ∠5 are supplementary. So, ∠6=180 - 112 = 68°.

Answer:

1. $x = 28^{\circ}$
2. $x = 65^{\circ}$
3. ∠1 = 68°
4. ∠2 = 112°
5. ∠3 = 90°
6. ∠4 = 90°
7. ∠5 = 112°
8. ∠6 = 68°