15. In AXYZ, XY = YZ and m/XYZ = 46°. 46° What is the measure of Z XZY? A. 23 В. 44° C. 46" D. 67' IN

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
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Question 15

### Geometry Problem: Isosceles Triangle Angle Calculation

**Problem Statement:**
In triangle ΔXYZ, it is given that sides XY and YZ are congruent (XY ≅ YZ). The measure of angle XYZ is provided as 46°.

**Question:**
What is the measure of ∠XZY?

**Solution Approach:**

1. **Identify Given Information:**
   - Triangle ΔXYZ with XY ≅ YZ.
   - m∠XYZ (angle XYZ) = 46°.

2. **Analyze Geometric Relationships:**
   - Since XY ≅ YZ, it is an isosceles triangle with YYZ and YYX as the legs.
   - In isosceles triangles, the base angles (angles opposite the equal sides) are congruent. Thus, ∠XYX = ∠XZY.
   - Use the angle sum property of triangles, which states that the sum of the interior angles of a triangle is 180°.

3. **Calculate the Measure:**
   - Let m∠XZY be denoted as x.
   - Therefore, m∠XYZ + m∠XYX + m∠XZY = 180°.
   - Substituting the known values: 46° + x + x = 180°.
   - Simplifying, we get: 46° + 2x = 180°.
   - 2x = 180° - 46°.
   - 2x = 134°.
   - x = 134° / 2.
   - x = 67°.

Thus, the measure of angle ∠XZY is **67°**.

**Answer Choices:**
- A. 23°
- B. 44°
- C. 46°
- D. 67° (Correct Answer, indicated by a blue check mark).

**Diagram:**
The image includes a diagram of an isosceles triangle ΔXYZ. Angles and sides are marked as follows:
- XY and YZ are marked as congruent (indicated by the hash marks on the sides).
- ∠XYZ is marked as 46°.

This problem involves understanding the properties of isosceles triangles and applying the angle sum property to determine the measure of the unknown angle.
Transcribed Image Text:### Geometry Problem: Isosceles Triangle Angle Calculation **Problem Statement:** In triangle ΔXYZ, it is given that sides XY and YZ are congruent (XY ≅ YZ). The measure of angle XYZ is provided as 46°. **Question:** What is the measure of ∠XZY? **Solution Approach:** 1. **Identify Given Information:** - Triangle ΔXYZ with XY ≅ YZ. - m∠XYZ (angle XYZ) = 46°. 2. **Analyze Geometric Relationships:** - Since XY ≅ YZ, it is an isosceles triangle with YYZ and YYX as the legs. - In isosceles triangles, the base angles (angles opposite the equal sides) are congruent. Thus, ∠XYX = ∠XZY. - Use the angle sum property of triangles, which states that the sum of the interior angles of a triangle is 180°. 3. **Calculate the Measure:** - Let m∠XZY be denoted as x. - Therefore, m∠XYZ + m∠XYX + m∠XZY = 180°. - Substituting the known values: 46° + x + x = 180°. - Simplifying, we get: 46° + 2x = 180°. - 2x = 180° - 46°. - 2x = 134°. - x = 134° / 2. - x = 67°. Thus, the measure of angle ∠XZY is **67°**. **Answer Choices:** - A. 23° - B. 44° - C. 46° - D. 67° (Correct Answer, indicated by a blue check mark). **Diagram:** The image includes a diagram of an isosceles triangle ΔXYZ. Angles and sides are marked as follows: - XY and YZ are marked as congruent (indicated by the hash marks on the sides). - ∠XYZ is marked as 46°. This problem involves understanding the properties of isosceles triangles and applying the angle sum property to determine the measure of the unknown angle.
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