Question

Given: AD and GC bisect each otherProve: angle GHA and angle CHD are congruent

Answer 1

Given the information from this diagram;

`AD\text{ and GC bisect each other}`

This means at point H, which is the point of intersection;

`\begin{gathered} CH=GH \\ \text{Also;} \\ AH=DH \end{gathered}`

This means, in triangle GHA and triangle CHD;

`\begin{gathered} CH\cong GH \\ AH\cong DH \\ \angle AHG=\angle CHD\text{ (opposite angles)} \\ \text{Therefore;} \\ \Delta GHA\cong\Delta CHD \end{gathered}`

To get a third piece of information about the two triangles, we would consider the third side of each triangle, which would be;

`AG\text{ and CD}`

Note that two sides of both triangles are congruent, and one angle is congruent in both triangles. Therefore, we can conclude that the third segment AG in triangle GHA is congruent to segment CD in triangle CHD.

Therefore;

ANSWER:

`AG,\text{ DC are Congruent}`

`\begin{gathered} \angle AHG,\angle CHD \\ \text{are Opposite Angles} \end{gathered}`
`\text{Side}-Angle-Side\text{ Postulate}`

Since we have two sides congruent and one angle congruent for both triangles, then we have enough information to show that the corresponding third side for both triangles are congruent.