Question
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)
Answers
GPT-4o mini
To prove that the two triangles are congruent using the Triangle Sum Theorem and the AAS (Angle-Angle-Side) congruency theorem, you can follow these steps:
### Step 1: Use the Triangle Sum Theorem
The Triangle Sum Theorem states that the sum of the angles in a triangle is always 180 degrees.
For triangle XYZ, we have:
- \(\angle X = 87^\circ\)
- \(\angle Z = 57^\circ\)
To find \(\angle Y\), we can use the Triangle Sum Theorem:
\[
\angle Y = 180^\circ - \angle X - \angle Z = 180^\circ - 87^\circ - 57^\circ = 36^\circ
\]
### Step 2: Compare Angles
Now we have:
- \(\angle Y = 36^\circ\)
For triangle BCD, we know:
- \(\angle D = 36^\circ\)
- \(\angle B = 57^\circ\)
Now we need to check the measures:
- \(\angle C\) can be found similarly:
\[
\angle C = 180^\circ - \angle D - \angle B = 180^\circ - 36^\circ - 57^\circ = 87^\circ
\]
So now we have:
- \(\angle B = 57^\circ\)
- \(\angle D = 36^\circ\)
- \(\angle C = 87^\circ\)
### Step 3: Establish Corresponding Angles
Now we can establish correspondences:
- \(\angle X = \angle C = 87^\circ\)
- \(\angle Z = \angle B = 57^\circ\)
- \(\angle Y = \angle D = 36^\circ\)
### Step 4: Use AAS Congruency Theorem
The AAS congruency theorem states that if two angles and a non-included side of one triangle are congruent to the two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
In this case:
- We have two angles from triangle XYZ (\(\angle X\) and \(\angle Z\)) that are congruent to two angles in triangle BCD (\(\angle C\) and \(\angle B\)).
- The side \(XY = 8\) (which corresponds to \(CD = 8\)) is congruent.
Since we have:
- \(\angle X = \angle C\)
- \(\angle Z = \angle B\)
- Side \(XY = CD\)
We can conclude that the triangles XYZ and BCD are congruent by the AAS theorem.
### Conclusion
Thus, we have used the Triangle Sum Theorem to find all angles in both triangles and identified corresponding angles. With two angles and the corresponding non-included side being equal, we have proven that the two triangles are congruent using the AAS theorem.
### Step 1: Use the Triangle Sum Theorem
The Triangle Sum Theorem states that the sum of the angles in a triangle is always 180 degrees.
For triangle XYZ, we have:
- \(\angle X = 87^\circ\)
- \(\angle Z = 57^\circ\)
To find \(\angle Y\), we can use the Triangle Sum Theorem:
\[
\angle Y = 180^\circ - \angle X - \angle Z = 180^\circ - 87^\circ - 57^\circ = 36^\circ
\]
### Step 2: Compare Angles
Now we have:
- \(\angle Y = 36^\circ\)
For triangle BCD, we know:
- \(\angle D = 36^\circ\)
- \(\angle B = 57^\circ\)
Now we need to check the measures:
- \(\angle C\) can be found similarly:
\[
\angle C = 180^\circ - \angle D - \angle B = 180^\circ - 36^\circ - 57^\circ = 87^\circ
\]
So now we have:
- \(\angle B = 57^\circ\)
- \(\angle D = 36^\circ\)
- \(\angle C = 87^\circ\)
### Step 3: Establish Corresponding Angles
Now we can establish correspondences:
- \(\angle X = \angle C = 87^\circ\)
- \(\angle Z = \angle B = 57^\circ\)
- \(\angle Y = \angle D = 36^\circ\)
### Step 4: Use AAS Congruency Theorem
The AAS congruency theorem states that if two angles and a non-included side of one triangle are congruent to the two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
In this case:
- We have two angles from triangle XYZ (\(\angle X\) and \(\angle Z\)) that are congruent to two angles in triangle BCD (\(\angle C\) and \(\angle B\)).
- The side \(XY = 8\) (which corresponds to \(CD = 8\)) is congruent.
Since we have:
- \(\angle X = \angle C\)
- \(\angle Z = \angle B\)
- Side \(XY = CD\)
We can conclude that the triangles XYZ and BCD are congruent by the AAS theorem.
### Conclusion
Thus, we have used the Triangle Sum Theorem to find all angles in both triangles and identified corresponding angles. With two angles and the corresponding non-included side being equal, we have proven that the two triangles are congruent using the AAS theorem.