Question
For the picture below: XY = 8, CD = 8. <X = 87 degrees, <Z=57 degrees, <D = 36 degrees and <B = 57 degrees.
Part 1. Explain how you can use the Triangle Sum theorem and the AAS congruency theorem to prove the two triangles congruent. (2pts)
Part 2. Write a correct congruency statement. You may use the = instead of the congruency symbol. (2pts)
(4 points)
Part 1. Explain how you can use the Triangle Sum theorem and the AAS congruency theorem to prove the two triangles congruent. (2pts)
Part 2. Write a correct congruency statement. You may use the = instead of the congruency symbol. (2pts)
(4 points)
Answers
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Answered by
GPT-4o mini
Let's break down the problem into the two parts requested:
### Part 1: Explanation of Congruence
To prove that the two triangles are congruent using the Triangle Sum Theorem and the AAS (Angle-Angle-Side) Congruency Theorem, follow these steps:
1. **Triangle Sum Theorem**: This theorem states that the sum of the angles in a triangle is always 180 degrees. We can use the known angles to find any missing angles in the triangles.
- For one triangle, we know two angles, \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- We can find the third angle \( \angle Y \) using the Triangle Sum Theorem:
\[
\angle Y = 180^\circ - \angle X - \angle Z = 180^\circ - 87^\circ - 57^\circ = 36^\circ
\]
- The third angle \( \angle Y \) is found to be \( 36^\circ \).
2. **Using AAS Congruency Theorem**: We now have enough information to apply the AAS Congruency Theorem. This theorem states that if two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then the two triangles are congruent.
- In Triangle XYZ: \( \angle X = 87^\circ \), \( \angle Z = 57^\circ \), and side \( YZ = 8 \) (since we established CD = 8).
- In Triangle CDZ: \( \angle D = 36^\circ \) (which we found), \( \angle B = 57^\circ\), and side \( CD = 8 \).
Since we have established:
- \( \angle X = \angle D = 87^\circ \)
- \( \angle Z = \angle B = 57^\circ \)
- Side \( YZ = CD = 8 \)
With two angles and the side included being equal, we conclude that the triangles are congruent by AAS.
### Part 2: Congruency Statement
From the triangles we analyzed, we can write the congruency statement as follows:
\[
\triangle XYZ = \triangle DBC
\]
Both parts of the answer highlight the importance of angle and side relationships in proving triangle congruency using geometric theorems.
### Part 1: Explanation of Congruence
To prove that the two triangles are congruent using the Triangle Sum Theorem and the AAS (Angle-Angle-Side) Congruency Theorem, follow these steps:
1. **Triangle Sum Theorem**: This theorem states that the sum of the angles in a triangle is always 180 degrees. We can use the known angles to find any missing angles in the triangles.
- For one triangle, we know two angles, \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- We can find the third angle \( \angle Y \) using the Triangle Sum Theorem:
\[
\angle Y = 180^\circ - \angle X - \angle Z = 180^\circ - 87^\circ - 57^\circ = 36^\circ
\]
- The third angle \( \angle Y \) is found to be \( 36^\circ \).
2. **Using AAS Congruency Theorem**: We now have enough information to apply the AAS Congruency Theorem. This theorem states that if two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then the two triangles are congruent.
- In Triangle XYZ: \( \angle X = 87^\circ \), \( \angle Z = 57^\circ \), and side \( YZ = 8 \) (since we established CD = 8).
- In Triangle CDZ: \( \angle D = 36^\circ \) (which we found), \( \angle B = 57^\circ\), and side \( CD = 8 \).
Since we have established:
- \( \angle X = \angle D = 87^\circ \)
- \( \angle Z = \angle B = 57^\circ \)
- Side \( YZ = CD = 8 \)
With two angles and the side included being equal, we conclude that the triangles are congruent by AAS.
### Part 2: Congruency Statement
From the triangles we analyzed, we can write the congruency statement as follows:
\[
\triangle XYZ = \triangle DBC
\]
Both parts of the answer highlight the importance of angle and side relationships in proving triangle congruency using geometric theorems.
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