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.