$$ riangle K A T $$ is congruent to $$ riangle X Y Z $$ In $$ \Delta Y Y Z, ~ $$ I $$ =45^{\circ} $$ and in $$ riangle K A T ~ $$ $$ A A= $$ $$ 78^{\circ} $$. Find the measure of $$ \measuredangle Z $$.

Question

$$ 	riangle K A T $$ is congruent to $$ 	riangle X Y Z $$ In $$ \Delta Y Y Z, ~ $$ I $$ =45^{\circ} $$ and in $$ 	riangle K A T ~ $$ $$ A A= $$ $$ 78^{\circ} $$. Find the measure of $$ \measuredangle Z $$.

UpStudy AI · Accepted Answer

To solve the problem, we need to use the properties of congruent triangles. Since $$ \triangle KAT $$ is congruent to $$ \triangle XYZ $$, the corresponding angles of these triangles are equal.

Given:
- In $$ \triangle XYZ $$, $$ \angle Y = 45^\circ $$.
- In $$ \triangle KAT $$, $$ \angle A = 78^\circ $$.

Since the triangles are congruent, we can set up the following relationships for the corresponding angles:
- $$ \angle K = \angle X $$
- $$ \angle A = \angle Y $$
- $$ \angle T = \angle Z $$

From the information provided:
- $$ \angle Y = 45^\circ $$ implies $$ \angle A = 45^\circ $$ (since $$ \angle A $$ corresponds to $$ \angle Y $$).
- $$ \angle A = 78^\circ $$ implies $$ \angle K = 78^\circ $$ (since $$ \angle K $$ corresponds to $$ \angle A $$).

Now, we can find $$ \angle Z $$ using the fact that the sum of the angles in a triangle is $$ 180^\circ $$.

In $$ \triangle XYZ $$:
$$
\angle X + \angle Y + \angle Z = 180^\circ
$$

Substituting the known values:
$$
\angle K + 45^\circ + \angle Z = 180^\circ
$$

Since $$ \angle K = 78^\circ $$:
$$
78^\circ + 45^\circ + \angle Z = 180^\circ
$$

Now, we can solve for $$ \angle Z $$:
$$
\angle Z = 180^\circ - 78^\circ - 45^\circ
$$

Calculating this gives:
$$
\angle Z = 180^\circ - 123^\circ = 57^\circ
$$

Thus, the measure of $$ \angle Z $$ is $$ 57^\circ $$.
The measure of $$ \angle Z $$ is $$ 57^\circ $$.

\( \triangle K A T \) is congruent to \( \triangle X Y Z \) In \( \Delta Y Y Z, ~ \) I \( =45^{\circ} \) and in \( \triangle K A T ~ \) \( A A= \) \( 78^{\circ} \). Find the measure of \( \measuredangle Z \).

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