We are asked to prove Theorem 6.23, which is a biconditional statement.
A trapezoid is isosceles
if and only if
its diagonals are congruent.
Let's draw a diagram, label the vertices and indicate the parallel bases of the trapezoid.
We will show the two directions separately.
If a Trapezoid Is Isosceles Then Its Diagonals Are Congruent
Let's compare the sides and angles of these triangles.
Congruence
Justification
QT≅RS
Given (legs of an isosceles trapezoid)
∠ TQR≅∠ SRQ
Theorem 6.21 (base angles of an isosceles trapezoid)
Since QR is a common side of triangles △ QRT and △ RQS, these triangles have two congruent sides and congruent included angles. According to the Side-Angle-Side (SAS) Congruence Postulate, the triangles are congruent.
△ QRT≅ △ RQS
Corresponding sides of congruent triangles are congruent.
QS≅RT
These are the diagonals of QRST, so we proved that the diagonals of an isosceles trapezoid are congruent. Let's summarize the steps above in a paragraph proof.
Completed Proof
2
&Given:&& QRST is a trapezoid
& && QR∥TS, QT≅RS
&Prove:&& QS≅RT
Proof:
Angles ∠ TQR and ∠ SRQ are base angles of an isosceles trapezoid, so according to Theorem 6.21 they are congruent. It is given that QT≅RT. Since QR is a common side of triangles △ QRT and △ RQS, according to the SAS Postulate the triangles are congruent. Diagonals QS and RT are corresponding sides of these congruent triangles, so they are congruent.
Let's prove now the converse of this statement.
If the Diagonals of a Trapezoid Are Congruent, Then the Trapezoid Is Isosceles
Let's draw a diagram, indicate the parallel bases and congruent diagonals, and extend the diagram with a segment parallel to one of the diagonals. We will find angles congruent to the angle at U and show that the triangles shaded on the diagram are congruent.
We will need to focus on different parts of the diagram at the different stages of the proof. Let's start with quadrilateral QRUS.
By construction, opposite sides of this quadrilateral are parallel, so QRUS is a parallelogram. According to Theorem 6.3 and Theorem 6.4, opposite sides and angles of a parallelogram are congruent. Let's list one congruent side pair and one congruent angle pair.
QS≅RU
∠ RUS≅∠ SQR
Since it is given that diagonal QS is also congruent to RT, we now know that triangle △ RTU is isosceles.
According to the Isosceles Triangle Theorem, the base angles are congruent.
∠ RUT≅∠ RTU
Let's focus now on the parallel sides of trapezoid QRST and think of the diagonals as transversals.
According to the Alternate Interior Angles Theorem, the angles formed by the diagonal and the parallel bases of the trapezoid are congruent.
∠ RQS≅∠ TSQ
∠ QRT≅∠ STR
We have four angles that are congruent to each other, so triangles △ QRO and △ STO are isosceles. Let's indicate the congruent sides on the diagram.
These congruent sides are also sides of triangles △ QOT and △ ROS.
The included angles between the corresponding congruent sides in these two triangles are vertical angles, so they are congruent.
∠ QOT≅∠ ROS
According to the Side-Angle-Side (SAS) Congruence Postulate, this means that the triangles are congruent.
△ QOT≅ △ ROS
Segments QT and RS are corresponding sides in these triangles, so they are also congruent. Since QT and RS are the legs of trapezoid QRST, this finishes the proof that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. Let's summarize the steps above in a paragraph proof.
Completed Proof
2
&Given:&& QRST is a trapezoid
& && QR∥TS, QS≅RT
&Prove:&& QT≅RS
Proof:
Let's extend the trapezoid with parallelogram QRUS. Opposite sides and angles of a parallelogram are congruent, so QS≅RU and ∠ RUS≅∠ SQR. Since it is given that diagonal QS is also congruent to RT, we now know that triangle △ RTU is isosceles, so its base angles, ∠ RUT and ∠ RTU are congruent. The angles formed by the diagonals and the bases of the trapezoid are alternate interior angles, and hence congruent. This gives us four congruent angles, ∠ RQS≅∠ QRT≅ RTQ≅ ∠ QST, and two isosceles triangles △ QRO and △ STO. The congruent sides of these triangles are corresponding sides in triangles △ QOT and △ ROS. The included angles between these congruent sides are vertical, so congruent. According to SAS this means that triangles △ QOT and △ ROS are congruent. The legs of the trapezoid, QT and RS, are corresponding sides in these congruent triangles, so they are congruent. This means that QRST is an isosceles trapezoid.
