Find angles with measure less than ∠7 using triangle angle inequality rules

Question

Select the angle(s) with measures that are less than m∠7. The triangle diagram has points X, T, W, V on a straight line, and point U above the line. Angles are labeled as ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, ∠TUV.

Accepted Answer

∠5, ∠6, ∠2, ∠1, ∠4

Answer

Step 1: Recall the exterior angle theorem for triangles: An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, so the exterior angle is greater than each of the non-adjacent interior angles. Step 2: Analyze ∠7: It is an exterior angle of △UTW. So m∠7 > m∠1 and m∠7 > m∠2 (non-adjacent interior angles of the triangle with exterior ∠7). Step 3: ∠1 is an exterior angle of △UWV, so m∠1 > m∠4 and m∠1 > m∠6. Since m∠7 > m∠1, this means m∠7 > m∠4 and m∠7 > m∠6. Step 4: ∠2 is an exterior angle of △UTU (or directly, ∠7 is also related to △UTV: ∠7 is part of the exterior angle for △U TV, and ∠5 is an interior angle of △UWV, and since ∠1 > ∠5 and ∠7 > ∠1, then m∠7 > m∠5. Step 5: Evaluate remaining angles: m∠3 is greater than m∠7 (as ∠3 is an exterior angle to the triangle containing ∠7), m∠TUV is a large angle containing ∠7, so it is larger than ∠7.