Question

quadrilateral qrst is an isosceles trapezoid and m∠s = 7b - 99°. what is the value of b? image of isosceles trapezoid qrst with angle t labeled 104°

Explanation:

Step1: Recall isosceles trapezoid angle property

In an isosceles trapezoid, consecutive angles between the bases are supplementary. Also, base angles are equal, and adjacent angles (like ∠T and ∠S) are supplementary? Wait, no, in trapezoid QRST, sides TQ and SR? Wait, looking at the diagram, ∠T is 104°, and ∠S is adjacent to ∠T. In a trapezoid, consecutive angles between the legs (if it's an isosceles trapezoid) – wait, actually, in an isosceles trapezoid, each pair of base angles is equal, and consecutive angles along a leg are supplementary. So if QRST is an isosceles trapezoid, then ∠T and ∠S – wait, maybe ∠T and ∠S are supplementary? Wait, no, let's check the diagram. The trapezoid has vertices T, S, R, Q. So sides TS and QR? Wait, maybe TS and QR are the legs? No, maybe TQ and SR are the legs. Wait, the angle at T is 104°, and angle at S is m∠S = 7b - 99°. In an isosceles trapezoid, consecutive angles between the bases are supplementary. Wait, maybe ∠T and ∠S are supplementary? Wait, no, if it's a trapezoid with bases TQ and SR, then angles at T and S would be adjacent to the leg TS. Wait, maybe I made a mistake. Wait, in an isosceles trapezoid, base angles are equal, and consecutive angles (along a leg) are supplementary. So if ∠T is 104°, then ∠S should be equal to ∠T? No, that can't be. Wait, no, maybe ∠T and ∠S are supplementary? Wait, no, let's think again. In a trapezoid, the sum of consecutive angles between the legs (the non-parallel sides) – wait, no, the parallel sides are called bases. So if QR and TS are the bases (parallel), then the legs are TQ and SR. Then, angles at T and S are adjacent to leg TS, so they are consecutive angles along leg TS, so they should be supplementary? Wait, no, consecutive angles between the bases (the parallel sides) are supplementary. So if QR || TS, then ∠Q + ∠T = 180°, ∠R + ∠S = 180°, and ∠T = ∠Q, ∠S = ∠R (in isosceles trapezoid). Wait, but in the diagram, ∠T is 104°, so maybe ∠S is equal to ∠T? No, that would mean they are equal, but then 7b - 99 = 104? Let's check that. Wait, maybe the problem is that ∠T and ∠S are equal? Wait, no, in an isosceles trapezoid, base angles are equal. So if the bases are QR and TS, then angles at Q and R are base angles, and angles at T and S are base angles. Wait, maybe I got the vertices wrong. Let's list the vertices in order: T, S, R, Q. So the sides are TS, SR, RQ, QT. So TS and RQ are the two bases (parallel), and SR and QT are the legs (non-parallel). Then, in an isosceles trapezoid, the base angles (angles at T and Q, angles at S and R) are equal. Wait, no, angles at T and S are adjacent to base TS, and angles at Q and R are adjacent to base RQ. So angles at T and S: if TS is a base, then angles at T (∠T) and S (∠S) are adjacent to leg QT and SR. Wait, maybe ∠T and ∠S are supplementary? Wait, no, in a trapezoid, consecutive angles between the legs (the non-parallel sides) are supplementary. Wait, I'm confused. Wait, the problem says it's an isosceles trapezoid, and m∠S = 7b - 99°, and ∠T is 104°. Maybe ∠T and ∠S are equal? Let's test that. If ∠T = ∠S, then 104 = 7b - 99. Then solve for b: 7b = 104 + 99 = 203, so b = 203 / 7 = 29. Wait, that works. Wait, maybe in the isosceles trapezoid, ∠T and ∠S are equal? Let's check the properties. In an isosceles trapezoid, each pair of base angles is equal. So if TS and RQ are the bases, then angles at T (∠T) and S (∠S) are base angles, so they should be equal. Yes, that makes sense. So ∠T = ∠S. So m∠T = 104°, so m∠S = 104°. Therefore, 7b - 99 = 104.

Step2: Solve for b

Set up the equation: 7b - 99 = 104.
Add 99 to both sides: 7b = 104 + 99 = 203.
Divide both sides by 7: b = 203 / 7 = 29.

Answer:

29