Question

a figure is shown. complete the statements. the congruent - complements theorem implies that if (mangle agf) is (60^{circ}), (mangle cgh) is. the vertical angles theorem implies that if (mangle igd) is, (mangle cgh) is (20^{circ}).

Explanation:

Step1: Recall congruent - complements theorem

The congruent - complements theorem states that if two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. $\angle AGF$ and $\angle CGH$ are both complementary to $\angle FGC$. Since $\angle AGB = 90^{\circ}$, if $m\angle AGF=60^{\circ}$, then $m\angle FGC = 90^{\circ}-60^{\circ}=30^{\circ}$. And since $\angle FGC+\angle CGH = 90^{\circ}$, then $m\angle CGH = 60^{\circ}$ because they are complements of the same angle $\angle FGC$.

Step2: Recall vertical - angles theorem

Vertical angles are congruent. $\angle IGD$ and $\angle CGH$ are vertical angles. If $m\angle CGH = 20^{\circ}$, then by the vertical - angles theorem, $m\angle IGD=20^{\circ}$.

Answer:

The congruent complements theorem implies that if $m\angle AGF$ is $60^{\circ}$, $m\angle CGH$ is $60^{\circ}$.
The vertical angles theorem implies that if $m\angle IGD$ is $20^{\circ}$, $m\angle CGH$ is $20^{\circ}$.