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**Problem Statement:** Given the picture below, what is the value of \( x \) and the measure of \( \angle B \)? **Diagram Description:** The diagram shows triangle \( ABC \) with: - \( \overline{AB} \) and \( \overline{BC} \) marked as congruent. - Angle \( \angle A \) is labeled as \( 4x - 18^\circ \). - Angle \( \angle C \) is labeled as \( 3x^\circ \). **Solution:** 1. **Calculate the value of \( x \):** Since \( \overline{AB} \cong \overline{BC} \), triangle \( ABC \) is isosceles, implying \( \angle A \cong \angle C \). Therefore, \( 4x - 18 = 3x \). Solving for \( x \): \[ 4x - 18^\circ = 3x^\circ \] \[ 4x - 3x = 18 \] \[ x = 18 \] 2. **Calculate the measure of \( \angle B \):** The measures of angles in any triangle sum up to \( 180^\circ \). \[ \angle A + \angle B + \angle C = 180^\circ \] Substituting the given expressions: \[ (4x - 18) + \angle B + 3x = 180 \] Substitute \( x = 18 \): \[ (4(18) - 18) + \angle B + 3(18) = 180 \] \[ (72 - 18) + \angle B + 54 = 180 \] \[ 54 + \angle B + 54 = 180 \] \[ 108 + \angle B = 180 \] \[ \angle B = 72^\circ \] **Final Answer:** \( x = 18 \) \( \angle B = 72^\circ \)

**Problem Statement:** Given the picture below, what is the value of \( x \) and the measure of \( \angle B \)? **Diagram Description:** The diagram shows triangle \( ABC \) with: - \( \overline{AB} \) and \( \overline{BC} \) marked as congruent. - Angle \( \angle A \) is labeled as \( 4x - 18^\circ \). - Angle \( \angle C \) is labeled as \( 3x^\circ \). **Solution:** 1. **Calculate the value of \( x \):** Since \( \overline{AB} \cong \overline{BC} \), triangle \( ABC \) is isosceles, implying \( \angle A \cong \angle C \). Therefore, \( 4x - 18 = 3x \). Solving for \( x \): \[ 4x - 18^\circ = 3x^\circ \] \[ 4x - 3x = 18 \] \[ x = 18 \] 2. **Calculate the measure of \( \angle B \):** The measures of angles in any triangle sum up to \( 180^\circ \). \[ \angle A + \angle B + \angle C = 180^\circ \] Substituting the given expressions: \[ (4x - 18) + \angle B + 3x = 180 \] Substitute \( x = 18 \): \[ (4(18) - 18) + \angle B + 3(18) = 180 \] \[ (72 - 18) + \angle B + 54 = 180 \] \[ 54 + \angle B + 54 = 180 \] \[ 108 + \angle B = 180 \] \[ \angle B = 72^\circ \] **Final Answer:** \( x = 18 \) \( \angle B = 72^\circ \)

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Problem Statement:** Given the picture below, what is the value of \( x \) and the measure of \( \angle B \)? **Diagram Description:** The diagram shows triangle \( ABC \) with: - \( \overline{AB} \) and \( \overline{BC} \) marked as congruent. - Angle \( \angle A \) is labeled as \( 4x - 18^\circ \). - Angle \( \angle C \) is labeled as \( 3x^\circ \). **Solution:** 1. **Calculate the value of \( x \):** Since \( \overline{AB} \cong \overline{BC} \), triangle \( ABC \) is isosceles, implying \( \angle A \cong \angle C \). Therefore, \( 4x - 18 = 3x \). Solving for \( x \): \[ 4x - 18^\circ = 3x^\circ \] \[ 4x - 3x = 18 \] \[ x = 18 \] 2. **Calculate the measure of \( \angle B \):** The measures of angles in any triangle sum up to \( 180^\circ \). \[ \angle A + \angle B + \angle C = 180^\circ \] Substituting the given expressions: \[ (4x - 18) + \angle B + 3x = 180 \] Substitute \( x = 18 \): \[ (4(18) - 18) + \angle B + 3(18) = 180 \] \[ (72 - 18) + \angle B + 54 = 180 \] \[ 54 + \angle B + 54 = 180 \] \[ 108 + \angle B = 180 \] \[ \angle B = 72^\circ \] **Final Answer:** \( x = 18 \) \( \angle B = 72^\circ \)
Transcribed Image Text:**Problem Statement:** Given the picture below, what is the value of \( x \) and the measure of \( \angle B \)? **Diagram Description:** The diagram shows triangle \( ABC \) with: - \( \overline{AB} \) and \( \overline{BC} \) marked as congruent. - Angle \( \angle A \) is labeled as \( 4x - 18^\circ \). - Angle \( \angle C \) is labeled as \( 3x^\circ \). **Solution:** 1. **Calculate the value of \( x \):** Since \( \overline{AB} \cong \overline{BC} \), triangle \( ABC \) is isosceles, implying \( \angle A \cong \angle C \). Therefore, \( 4x - 18 = 3x \). Solving for \( x \): \[ 4x - 18^\circ = 3x^\circ \] \[ 4x - 3x = 18 \] \[ x = 18 \] 2. **Calculate the measure of \( \angle B \):** The measures of angles in any triangle sum up to \( 180^\circ \). \[ \angle A + \angle B + \angle C = 180^\circ \] Substituting the given expressions: \[ (4x - 18) + \angle B + 3x = 180 \] Substitute \( x = 18 \): \[ (4(18) - 18) + \angle B + 3(18) = 180 \] \[ (72 - 18) + \angle B + 54 = 180 \] \[ 54 + \angle B + 54 = 180 \] \[ 108 + \angle B = 180 \] \[ \angle B = 72^\circ \] **Final Answer:** \( x = 18 \) \( \angle B = 72^\circ \)
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