find the measure of angle 4\n90\n22\n136\n44

Answer

Explanation:

Step1: Identify the shape type

The figure is a parallelogram (opposite sides congruent, marked with ticks). In a parallelogram, consecutive angles are supplementary.

Step2: Find adjacent angle

Let the given $68^\circ$ angle be $\angle ABC$. Its consecutive angle $\angle BAD = 180^\circ - 68^\circ = 112^\circ$.

Step3: Use diagonal angle property

The diagonal bisects $\angle BAD$, so $\angle 1 = \frac{112^\circ}{2} = 56^\circ$.

Step4: Use triangle angle sum

In the triangle with $\angle 1$, $68^\circ$, and $\angle 4$: $\angle 4 = 180^\circ - 68^\circ - 56^\circ = 56^\circ$. Wait, correct: alternate approach: the $68^\circ$ angle and angle 4 form a triangle with the bisected angle. Wait, no: the $68^\circ$ angle, angle 1 (half of 112=56), so angle 4 = 180 - 68 - 56 = 56? No, wait, no: the $68^\circ$ angle is in a triangle with angle 4 and angle 1. Wait, no, actually, the $68^\circ$ angle and angle 4 are vertical? No, wait, the parallelogram's diagonal creates a triangle where the $68^\circ$ angle, angle 1 (which is equal to the angle opposite the $68^\circ$? No, wait, correct: in parallelogram, opposite angles are equal, so the angle opposite $68^\circ$ is $68^\circ$, the other two angles are $112^\circ$. The diagonal splits the $112^\circ$ angle into two $56^\circ$ angles. Then, in the triangle containing angle 4, we have $68^\circ$, $56^\circ$, so angle 4 = 180 - 68 - 56 = 56? No, that's not an option. Wait, no: angle 4 is supplementary to the angle adjacent to $68^\circ$? No, wait, angle 4 and the $68^\circ$ angle are same-side interior? No, wait, the $68^\circ$ angle and angle 4 form a linear pair with the bisected angle? Wait, no, alternative: angle 4 is equal to 180 - 68 - (180-136)? No, 136 is an option. Wait, angle 4 and the $68^\circ$ angle are supplementary? No, 180-68=112, not an option. Wait, wait: the figure has angle 4 and the $68^\circ$ angle, with the sides marked as congruent, so the triangle is isosceles. Wait, the two sides with ticks are congruent, so the triangle with $68^\circ$ angle is isosceles, so the base angles are equal. Wait, no, the $68^\circ$ angle is the vertex angle, so the base angles are $\frac{180-68}{2}=56$, then angle 4 is supplementary to 56? No, 180-56=124, not an option. Wait, no, angle 4 is vertical to the angle that's supplementary to 68? No, 180-68=112, not an option. Wait, wait, the options are 90,22,136,44. Oh! Wait, the $68^\circ$ angle is angle 4's adjacent angle in a triangle where the other angle is 44? No, 68+44+68=180? No. Wait, 180-68=112, half of that is 56, 180-56-82? No. Wait, maybe the $68^\circ$ angle is angle 3? No, the question is angle 4. Wait, angle 4 and angle 2 are vertical? No, angle 2 and angle 4 are vertical? No, angle 3 and angle 1 are vertical. Wait, the $68^\circ$ angle is angle 4's exterior angle? So angle 4 = 180-68-68? No, 44! Oh! Wait, if the triangle is isosceles with two angles equal to 68? No, 68+68+44=180. Yes! So angle 4 is 44. Because the two sides with ticks are congruent, so the base angles are 68, so angle 4 = 180 - 68 - 68 = 44. That's an option.

Step1: Recognize isosceles triangle

The triangle with $68^\circ$ has congruent sides (ticks), so it is isosceles, with two equal base angles of $68^\circ$.

Step2: Calculate angle 4

Use triangle angle sum: $\angle 4 = 180^\circ - 68^\circ - 68^\circ$ $\angle 4 = 44^\circ$