Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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![**Title: Understanding Isosceles Triangles**
**Topic: Using the Properties of Isosceles Triangles to Calculate Angles**
**Example Problem:**
**Instruction:**
Use the properties of isosceles and other triangles to find the indicated angle.
**Problem 11: Find \( m\angle ACB \)**
**Diagram Description:**
The diagram illustrates triangle \( \triangle ABC \) with an isosceles triangle \( \triangle ADC \).
- Vertex \( A \) is the common point of the two triangles.
- \( \angle ADC \) is marked as 45°.
- Side \( AD \) is congruent to side \( AC \), which means \(\triangle ADC\) is isosceles.
- Point \( B \) is connected to \( C \), forming segment \( BC \).
**Solution Explanation:**
1. **Identify the Isosceles Triangle:**
Since \( AD = AC \), triangle \( \triangle ADC \) is isosceles. This implies that angles \( \angle DAC \) and \( \angle ACD \) are equal.
2. **Use Known Angles:**
We know \( \angle ADC = 45^\circ \).
3. **Calculate \( \angle DAC \) and \( \angle ACD \):**
In an isosceles triangle, the sum of the angles is always 180°. Therefore,
\[
\angle DAC + \angle ACD + \angle ADC = 180^\circ
\]
\[
\angle DAC + \angle DAC + 45^\circ = 180^\circ
\]
\[
2(\angle DAC) = 135^\circ
\]
\[
\angle DAC = \angle ACD = 67.5^\circ
\]
4. **Conclusion:**
\( m\angle ACB = \angle ACD = 67.5^\circ \).
This exercise demonstrates how to utilize the properties of isosceles triangles to determine unknown angles, a crucial skill in geometry.](https://content.bartleby.com/qna-images/question/58eb46de-95f3-41c1-9329-93ec514644b2/24386718-82af-438f-98a2-92181df2eea8/3x2o1ot_processed.jpeg)
Transcribed Image Text:**Title: Understanding Isosceles Triangles**
**Topic: Using the Properties of Isosceles Triangles to Calculate Angles**
**Example Problem:**
**Instruction:**
Use the properties of isosceles and other triangles to find the indicated angle.
**Problem 11: Find \( m\angle ACB \)**
**Diagram Description:**
The diagram illustrates triangle \( \triangle ABC \) with an isosceles triangle \( \triangle ADC \).
- Vertex \( A \) is the common point of the two triangles.
- \( \angle ADC \) is marked as 45°.
- Side \( AD \) is congruent to side \( AC \), which means \(\triangle ADC\) is isosceles.
- Point \( B \) is connected to \( C \), forming segment \( BC \).
**Solution Explanation:**
1. **Identify the Isosceles Triangle:**
Since \( AD = AC \), triangle \( \triangle ADC \) is isosceles. This implies that angles \( \angle DAC \) and \( \angle ACD \) are equal.
2. **Use Known Angles:**
We know \( \angle ADC = 45^\circ \).
3. **Calculate \( \angle DAC \) and \( \angle ACD \):**
In an isosceles triangle, the sum of the angles is always 180°. Therefore,
\[
\angle DAC + \angle ACD + \angle ADC = 180^\circ
\]
\[
\angle DAC + \angle DAC + 45^\circ = 180^\circ
\]
\[
2(\angle DAC) = 135^\circ
\]
\[
\angle DAC = \angle ACD = 67.5^\circ
\]
4. **Conclusion:**
\( m\angle ACB = \angle ACD = 67.5^\circ \).
This exercise demonstrates how to utilize the properties of isosceles triangles to determine unknown angles, a crucial skill in geometry.
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