Question

5.
find the measures of each of the indicated angles.
$\angle 1$=
$\angle 2$=
$\angle 3$=
$\angle 4$=
$\angle 5$=

Explanation:

Step1: Find ∠2

Adjacent angles on a straight line sum to \(180^\circ\). Given the angle adjacent to ∠2 is \(156^\circ\), so \(\angle 2 = 180^\circ - 156^\circ = 24^\circ\).

Step2: Find ∠3

Adjacent angles on a straight line sum to \(180^\circ\). Given the angle adjacent to ∠3 is \(75^\circ\), so \(\angle 3 = 180^\circ - 75^\circ = 105^\circ\)? Wait, no, wait. Wait, the triangle has midlines? Wait, the original triangle and the smaller triangle: since the sides are marked as equal (midsegments), so the smaller triangle is similar? Wait, no, let's re - check. Wait, the angle adjacent to ∠2 is \(156^\circ\), so linear pair: \(\angle 2=180 - 156 = 24^\circ\). Then, in the smaller triangle, angles sum to \(180^\circ\). So \(\angle 1=180-(75 + 24)=81^\circ\). Then, for ∠3: since the sides are congruent (marked with ticks), the triangle is isosceles? Wait, the lower triangle: the angle adjacent to ∠3 is \(75^\circ\), but wait, maybe alternate interior angles? Wait, the line is a midline, so it's parallel to the base. So ∠4 should be equal to the angle above it (75°) because of alternate interior angles. ∠5 should be equal to ∠2 (24°) because of alternate interior angles. ∠3: since the triangle with ∠1, 75°, 24°: ∠1 = 81°, and ∠3: since the sides are congruent, maybe the triangle is isosceles? Wait, no, let's use the fact that the sum of angles in a triangle is \(180^\circ\). For the smaller triangle: \(\angle 1+\angle 2 + 75^\circ=180^\circ\), so \(\angle 1=180-(75 + 24)=81^\circ\). Then, ∠3: since the line is a midline, the triangle is similar? Wait, no, the sides are marked as equal (the ticks), so the segments are equal. So the triangle with ∠3, ∠4, ∠5: ∠4 = 75° (alternate interior with the 75° angle), ∠5 = 24° (alternate interior with ∠2), so ∠3=180-(75 + 24)=81°.

Step3: Find ∠4

Since the line is parallel to the base (midline theorem, as the sides are divided into equal segments), alternate interior angles are equal. So ∠4 is equal to the 75° angle, so \(\angle 4 = 75^\circ\).

Step4: Find ∠5

Similarly, alternate interior angles: ∠5 is equal to ∠2, so \(\angle 5=24^\circ\).

Step5: Verify ∠1

In the smaller triangle, sum of angles: \(75^\circ+\angle 2+\angle 1 = 180^\circ\). Substitute \(\angle 2 = 24^\circ\), so \(\angle 1=180-(75 + 24)=81^\circ\).

Step6: Verify ∠3

In the larger triangle, sum of angles: \(\angle 3+\angle 4+\angle 5=180^\circ\). Substitute \(\angle 4 = 75^\circ\) and \(\angle 5 = 24^\circ\), so \(\angle 3=180-(75 + 24)=81^\circ\).

Answer:

\(\angle 1 = 81^\circ\), \(\angle 2=24^\circ\), \(\angle 3 = 81^\circ\), \(\angle 4=75^\circ\), \(\angle 5 = 24^\circ\)