On the kite, vertex A at the top, vertex B at the right, vertex C at the bottom, and vertex D at the left. Side A B is marked congruent to side A D. Side D C is marked congruent to side B C. Diagonal A C and B D are drawn.<br/>Angle D A C is 39 degrees.
Find
m
∠
1
and
m
∠
3
in the kite.
1 / 2
Since side AB is congruent to side AD, angle ADB must be congruent to angle ABD (using the kite property that opposite angles are congruent). Therefore, angle ADB is also 39 degrees.
Using the same logic, we know that angle BCD and angle BAC are congruent, both measuring x degrees (let's call this angle B). Therefore, angle ABC is also x degrees.
Since the sum of angles in a triangle is 180 degrees, we can set up the following equation for triangle ABC:
x + x + 39 = 180
Simplifying, we get:
2x + 39 = 180
2x = 141
x = 70.5
Therefore, angle BAC and angle BCD both measure 70.5 degrees, making angle 1 equal to:
180 - 2(70.5) = 39 degrees
Similarly, angle 3 is the sum of angles ABD and BCD:
39 + 70.5 = 109.5 degrees