Question

for #34 – 35 use the diagram at right.

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.
(diagram: an isosceles triangle with two sides labeled 10, vertex angle 36°, and a segment from the base vertex to the extended base forming angles w and h)

Explanation:

Step1: Identify Triangle Type

The triangle has two sides of length 10, so it's isosceles. By the Base Angle Theorem, the base angles are equal. One base angle is \(36^\circ\), so the other base angle (angle \(w\)) is also \(36^\circ\)? Wait, no—wait, the angle at the top is \(36^\circ\), and the two equal sides are the legs? Wait, no, the diagram shows a triangle with two sides 10, so the base angles (opposite the equal sides) are equal. Wait, actually, the angle at the vertex is \(36^\circ\), and the two equal sides are the sides adjacent to the \(36^\circ\) angle? Wait, no, let's re-examine. The triangle has two sides of length 10, so it's isosceles with those two sides as the equal sides. Therefore, the base angles (the angles opposite the equal sides) are equal? Wait, no, the equal sides are the legs, so the base angles (the angles at the base) are equal. Wait, the angle given is \(36^\circ\) at the vertex. Then, using the Triangle Sum Theorem, the sum of angles in a triangle is \(180^\circ\). So the two base angles (let's call them \(w\)): \(36^\circ + w + w = 180^\circ\)? Wait, no, maybe I got the angle wrong. Wait, the diagram: there's a triangle with two sides 10, and an angle of \(36^\circ\) at the left vertex. Then, the other two angles: one is \(w\) (the angle at the base, adjacent to the extended side), and the other is... Wait, no, the triangle is isosceles with two sides 10, so the base angles (the angles opposite the equal sides) are equal. Wait, the equal sides are the two sides of length 10, so the angles opposite them are equal. Wait, the angle at the left is \(36^\circ\), so the other two angles: let's call them \(x\) and \(x\). Then \(36 + 2x = 180\), so \(2x = 144\), \(x = 72\). Wait, but then the angle \(w\) is adjacent to the extended side, forming a linear pair. So linear pair postulate: \(w + x = 180\)? Wait, no, maybe the angle \(w\) is the base angle? Wait, the options: A. 36, Base Angle Thm; B. 72, Base Angle Thm; C. 144, Triangle Sum Thm; D. 108, Linear Pair Post. Wait, let's re-express.

Wait, the triangle is isosceles with two sides 10, so the base angles (the angles opposite the equal sides) are equal. The vertex angle is \(36^\circ\), so the base angles (each) are \(\frac{180 - 36}{2} = 72^\circ\). But then, the angle \(w\) is adjacent to one of the base angles, forming a linear pair? Wait, no, the diagram shows a triangle with a side extended, forming an angle \(h\) and \(w\). Wait, maybe the angle \(w\) is the vertex angle? No, the options: A says 36, Base Angle Thm. Wait, Base Angle Theorem states that in an isosceles triangle, the angles opposite the equal sides are equal. So if the two sides are 10, the angles opposite them are equal. So if one angle is 36, the other is 36? But then the third angle would be 180 - 36 - 36 = 108. But that doesn't match. Wait, maybe the angle given is a base angle. So if the base angles are 36, then the vertex angle is 180 - 36 - 36 = 108. Then, the angle \(w\) is the vertex angle? No, the options: A. 36, Base Angle Thm; B. 72, Base Angle Thm; C. 144, Triangle Sum Thm; D. 108, Linear Pair Post.

Wait, let's look at the diagram again. The triangle has two sides of length 10, so it's isosceles. The angle at the left is 36 degrees. Then, the other two angles: one is \(w\) (at the base, adjacent to the extended side), and the other is... Wait, maybe the angle \(w\) is equal to 36 degrees by the Base Angle Theorem? Wait, Base Angle Theorem: if two sides are equal, the angles opposite are equal. So if the two sides of length 10 are the legs, then the angles opposite them (the base angles) are equal. Wait, the angle at the left is 36 degrees, which is a base angle, so the other base angle (angle \(w\)) is also 36 degrees. Then, the vertex angle is 180 - 36 - 36 = 108 degrees. But then, the angle \(h\) is adjacent to the vertex angle, forming a linear pair, so \(h = 180 - 108 = 72\). But the question is about \(w\). So if \(w\) is the base angle, equal to 36 degrees by the Base Angle Theorem, then option A. But wait, maybe I got the angle wrong. Wait, the two sides of length 10: the angles opposite them are equal. So if the angle at the left is 36 degrees, and it's opposite a side of length 10? No, the two sides of length 10 are adjacent to the 36-degree angle? Wait, no, in a triangle, the sides opposite the angles. So if two sides are equal, the angles opposite are equal. So if sides of length 10 are opposite angles of measure \(w\) and 36 degrees, then \(w = 36\) by Base Angle Theorem. Then, the third angle is 180 - 36 - 36 = 108. Then, the angle \(h\) is adjacent to the 108-degree angle, forming a linear pair, so \(h = 180 - 108 = 72\). But the question is about \(w\), so \(w = 36\) by Base Angle Theorem. So option A.

Wait, but let's check the options again. Option A: 36, Base Angle Thm. Option B: 72, Base Angle Thm. Option C: 144, Triangle Sum Thm. Option D: 108, Linear Pair Post.

Wait, maybe the angle given is the vertex angle. So vertex angle is 36 degrees, then the base angles are (180 - 36)/2 = 72 degrees. So \(w = 72\) by Base Angle Theorem. Then option B.

Ah, here's the confusion: which angle is the vertex angle? The diagram shows a triangle with two sides of length 10, so the vertex angle is the angle between the two equal sides. So the two equal sides are the legs, and the vertex angle is 36 degrees. Then the base angles (the angles opposite the equal sides) are (180 - 36)/2 = 72 degrees. So \(w = 72\) by Base Angle Theorem. So option B.

Yes, that makes sense. The Base Angle Theorem states that in an isosceles triangle, the angles opposite the equal sides are equal. So if the two equal sides are of length 10, the angles opposite them (the base angles) are equal. The vertex angle (between the two equal sides) is 36 degrees. Then, the base angles (each) are (180 - 36)/2 = 72 degrees. So \(w = 72\) by Base Angle Theorem. So option B.

Step2: Verify with Theorems

- **Base Angle Theorem**: Applies to isosceles triangles, stating that angles opposite equal sides are equal.
- **Triangle Sum Theorem**: Sum of angles in a triangle is \(180^\circ\). So \(36^\circ + w + w = 180^\circ\) (since two angles are equal, \(w\)). Solving: \(2w = 180 - 36 = 144\), so \(w = 72^\circ\). This confirms \(w = 72^\circ\) using the Base Angle Theorem (to identify the equal angles) and Triangle Sum Theorem (to calculate). But the option attributes it to Base Angle Theorem, so B is correct.

Answer:

B. 72, Base Angle Thm.