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)
Answers
GPT-4o mini
To prove that the two triangles are congruent, we can use the Triangle Sum theorem and the AAS (Angle-Angle-Side) congruency theorem as follows:
**Part 1: Explanation using Triangle Sum Theorem and AAS Congruency Theorem**
1. **Triangle Sum Theorem**: This theorem states that the sum of the interior angles of any triangle is always 180 degrees. For Triangle XYZ, we know that:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
To find the third angle \( \angle Y \), we can use the Triangle Sum theorem:
\[
\angle Y = 180^\circ - (\angle X + \angle Z) = 180^\circ - (87^\circ + 57^\circ) = 180^\circ - 144^\circ = 36^\circ
\]
So, \( \angle Y = 36^\circ \).
2. Now, we have the angles:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
- \( \angle Y = 36^\circ \)
For Triangle BCD, we already have:
- \( \angle D = 36^\circ \)
- \( \angle B = 57^\circ \)
To find \( \angle C \), we again use the Triangle Sum theorem:
\[
\angle C = 180^\circ - (\angle B + \angle D) = 180^\circ - (57^\circ + 36^\circ) = 180^\circ - 93^\circ = 87^\circ
\]
Thus, we have the angles:
- \( \angle C = 87^\circ \)
- \( \angle B = 57^\circ \)
- \( \angle D = 36^\circ \)
3. **Using AAS Congruency Theorem**: We now see that both triangles share two angles and a non-included side:
- For Triangle XYZ:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
- Side \( XY = 8 \)
- For Triangle BCD:
- \( \angle C = 87^\circ \)
- \( \angle B = 57^\circ \)
- Side \( CD = 8 \)
Since we have two angles and the side between those angles are equal in both triangles, we can conclude using the AAS theorem that Triangle XYZ is congruent to Triangle BCD.
**Part 2: Congruency Statement**
The congruency statement can be written as:
\[
\triangle XYZ = \triangle BCD
\]
**Summary:** We used the Triangle Sum theorem to find the missing angles in both triangles, and then applied the AAS congruency theorem to conclude that the two triangles are congruent.
**Part 1: Explanation using Triangle Sum Theorem and AAS Congruency Theorem**
1. **Triangle Sum Theorem**: This theorem states that the sum of the interior angles of any triangle is always 180 degrees. For Triangle XYZ, we know that:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
To find the third angle \( \angle Y \), we can use the Triangle Sum theorem:
\[
\angle Y = 180^\circ - (\angle X + \angle Z) = 180^\circ - (87^\circ + 57^\circ) = 180^\circ - 144^\circ = 36^\circ
\]
So, \( \angle Y = 36^\circ \).
2. Now, we have the angles:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
- \( \angle Y = 36^\circ \)
For Triangle BCD, we already have:
- \( \angle D = 36^\circ \)
- \( \angle B = 57^\circ \)
To find \( \angle C \), we again use the Triangle Sum theorem:
\[
\angle C = 180^\circ - (\angle B + \angle D) = 180^\circ - (57^\circ + 36^\circ) = 180^\circ - 93^\circ = 87^\circ
\]
Thus, we have the angles:
- \( \angle C = 87^\circ \)
- \( \angle B = 57^\circ \)
- \( \angle D = 36^\circ \)
3. **Using AAS Congruency Theorem**: We now see that both triangles share two angles and a non-included side:
- For Triangle XYZ:
- \( \angle X = 87^\circ \)
- \( \angle Z = 57^\circ \)
- Side \( XY = 8 \)
- For Triangle BCD:
- \( \angle C = 87^\circ \)
- \( \angle B = 57^\circ \)
- Side \( CD = 8 \)
Since we have two angles and the side between those angles are equal in both triangles, we can conclude using the AAS theorem that Triangle XYZ is congruent to Triangle BCD.
**Part 2: Congruency Statement**
The congruency statement can be written as:
\[
\triangle XYZ = \triangle BCD
\]
**Summary:** We used the Triangle Sum theorem to find the missing angles in both triangles, and then applied the AAS congruency theorem to conclude that the two triangles are congruent.
make it short
GPT-4o mini
Sure! Here’s a more concise version:
**Part 1: Explanation**
1. **Triangle Sum Theorem**: In Triangle XYZ, you know:
- \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- Calculate \( \angle Y \):
\[
\angle Y = 180^\circ - (87^\circ + 57^\circ) = 36^\circ.
\]
For Triangle BCD:
- \( \angle D = 36^\circ \) and \( \angle B = 57^\circ \).
- Calculate \( \angle C \):
\[
\angle C = 180^\circ - (57^\circ + 36^\circ) = 87^\circ.
\]
2. We now have:
- Triangle XYZ: \( \angle X = 87^\circ, \angle Y = 36^\circ, \angle Z = 57^\circ \).
- Triangle BCD: \( \angle B = 57^\circ, \angle C = 87^\circ, \angle D = 36^\circ \).
3. By the AAS congruency theorem (two angles and a non-included side), the triangles are congruent.
**Part 2: Congruency Statement**
\[
\triangle XYZ = \triangle BCD
\]
**Part 1: Explanation**
1. **Triangle Sum Theorem**: In Triangle XYZ, you know:
- \( \angle X = 87^\circ \) and \( \angle Z = 57^\circ \).
- Calculate \( \angle Y \):
\[
\angle Y = 180^\circ - (87^\circ + 57^\circ) = 36^\circ.
\]
For Triangle BCD:
- \( \angle D = 36^\circ \) and \( \angle B = 57^\circ \).
- Calculate \( \angle C \):
\[
\angle C = 180^\circ - (57^\circ + 36^\circ) = 87^\circ.
\]
2. We now have:
- Triangle XYZ: \( \angle X = 87^\circ, \angle Y = 36^\circ, \angle Z = 57^\circ \).
- Triangle BCD: \( \angle B = 57^\circ, \angle C = 87^\circ, \angle D = 36^\circ \).
3. By the AAS congruency theorem (two angles and a non-included side), the triangles are congruent.
**Part 2: Congruency Statement**
\[
\triangle XYZ = \triangle BCD
\]
Write a correct congruency statement. You may use the = instead of the congruency symbol.
GPT-4o mini
A correct congruency statement using "=" instead of the congruency symbol is:
\[
\triangle XYZ = \triangle BCD
\]
This indicates that Triangle XYZ is congruent to Triangle BCD.
\[
\triangle XYZ = \triangle BCD
\]
This indicates that Triangle XYZ is congruent to Triangle BCD.