Question

1. find the value of w and identify the theorem or postulate used.

a. 36°, base angle thm.
b. 72°, base angle thm.
c. 144°, triangle sum thm.
d. 108°, linear pair post.

1. find the value of h and identify the theorem or postulate used.

a. 36°, base angle thm.
b. 72°, base angle thm.
c. 144°, triangle sum thm.
d. 108°, linear pair post.
(there is a triangle with two sides labeled 10 and one angle labeled 36° at the left vertex, and an exterior angle related to angle h at the right vertex.)

Explanation:

Step1: Analyze the triangle (isosceles)

The triangle has two sides of length 10, so it's isosceles. By the Base Angle Theorem, the base angles are equal. One angle is \(36^\circ\), so the other base angle \(w\) is also \(36^\circ\)? Wait, no—wait, the vertex angle? Wait, no, the two equal sides are the legs, so the base angles are equal. Wait, the given angle is \(36^\circ\), so the other base angle (wait, no, maybe the vertex angle is \(36^\circ\), and the two base angles are equal. Let's use Triangle Sum Theorem: sum of angles in a triangle is \(180^\circ\). Wait, no, first, for angle \(w\): since the triangle is isosceles with sides 10, 10, so the base angles (opposite the equal sides) are equal. Wait, the angle given is \(36^\circ\), so if that's a base angle, then the other base angle is also \(36^\circ\), but then the vertex angle would be \(180 - 36 - 36 = 108^\circ\). But wait, the problem is about angle \(w\) and \(h\). Wait, maybe the first part (problem 34) is finding \(w\). Let's re-express:

For angle \(w\): The triangle is isosceles (two sides 10), so by Base Angle Theorem, the angles opposite the equal sides are equal. Wait, the angle given is \(36^\circ\), so if that's one base angle, then the other base angle (wait, no, maybe the vertex angle is \(36^\circ\), and the two base angles are equal. Wait, let's calculate:

Sum of angles in a triangle: \(180^\circ\). Let the two equal angles be \(w\) (since the sides are equal, the angles opposite are equal). Wait, the given angle is \(36^\circ\), so if that's a base angle, then \(w = 36^\circ\)? No, that can't be. Wait, maybe the given angle is the vertex angle. Wait, no, the diagram shows one angle as \(36^\circ\), and two sides 10. So the two angles opposite the 10s are equal. So if one angle is \(36^\circ\), then the other angle (opposite the other 10) is also \(36^\circ\), so the third angle (vertex angle) is \(180 - 36 - 36 = 108^\circ\). But wait, the problem is about angle \(w\) and the theorem. Wait, maybe for angle \(w\): since the triangle is isosceles, by Base Angle Theorem, the two base angles are equal. Wait, maybe the given angle is \(36^\circ\), so the other base angle (wait, no, maybe \(w\) is the base angle. Wait, let's check the options. Option B is \(72^\circ\), Base Angle Thm? Wait, maybe I made a mistake. Wait, no—wait, the angle adjacent to \(w\) is part of a linear pair? Wait, no, first, for angle \(w\):

Wait, the triangle has two sides of 10, so it's isosceles. The angle given is \(36^\circ\), so the two equal angles (base angles) are equal. Wait, maybe the vertex angle is \(36^\circ\), so the two base angles are \((180 - 36)/2 = 72^\circ\). Ah! That makes sense. So the vertex angle is \(36^\circ\), so the two base angles (opposite the equal sides) are equal, so each is \((180 - 36)/2 = 72^\circ\). So \(w = 72^\circ\), using Base Angle Theorem (since the triangle is isosceles, base angles are equal) and Triangle Sum? Wait, no, Base Angle Theorem says that if two sides are equal, the angles opposite are equal. So if the two sides are 10, the angles opposite (the base angles) are equal. So let the vertex angle be \(36^\circ\), then the two base angles (each \(w\)) satisfy \(2w + 36 = 180\), so \(2w = 144\), \(w = 72^\circ\). So that's using Base Angle Theorem (since the triangle is isosceles, base angles are equal) and solving. So for problem 34, the value of \(w\) is \(72^\circ\), theorem is Base Angle Thm, so option B.

For problem 35, angle \(h\) is adjacent to \(w\) forming a linear pair, so by Linear Pair Postulate, they are supplementary (sum to \(180^\circ\)). So \(h = 180 - w = 180 - 72 = 108^\circ\), using Linear Pair Postulate. So for \(h\), the value is \(108^\circ\), theorem is Linear Pair Postulate, so option D.

But let's focus on the first part (assuming problem 34: find \(w\)):

Step1: Identify triangle type (isosceles)

The triangle has two sides of length 10, so it's isosceles. By Base Angle Theorem, the angles opposite the equal sides are equal.

Step2: Calculate angle \(w\)

Let the vertex angle be \(36^\circ\). Then the two base angles (equal) are \(w\). Sum of angles: \(36 + w + w = 180\). Solve: \(2w = 180 - 36 = 144\), so \(w = 72^\circ\). This uses the Base Angle Theorem (to know the two base angles are equal) and Triangle Sum Theorem (to set up the equation). Wait, but the options for problem 34: option B is \(72^\circ\), Base Angle Thm. So that's the answer for \(w\).

For \(h\) (problem 35): \(h\) and \(w\) form a linear pair, so \(h + w = 180\). Thus, \(h = 180 - 72 = 108^\circ\), using Linear Pair Postulate. So option D for \(h\).

Answer:

For angle \(w\) (problem 34): \(\boldsymbol{72^\circ}\), Base Angle Thm (option B).
For angle \(h\) (problem 35): \(\boldsymbol{108^\circ}\), Linear Pair Postulate (option D).

(Assuming the questions are: 34. Find \(w\): B. \(72^\circ\), Base Angle Thm; 35. Find \(h\): D. \(108^\circ\), Linear Pair Postulate)