Given: Kite HJKL with diagonal Given: with diagonals

The following theorem is true for rectangles but not for parallelograms in general. All angles of a rectangle are right angles. COROLLARY 4.3.1 The diagonals of a rectangle are congruent. THEOREM 4.3.2 All sides of a square are congruent. COROLLARY 4.3.3 Reminder A rectangle is a parallelogram. Thus, it has all the properties of a parallelogram plus some properties of its own. NOTE: To follow the flow of the proof in Example 1, it may be best to draw triangles NMQ and PQM of Figure 4.21 separately. EXAMPLE 1 Complete a proof of Theorem 4.3.2. GIVEN: Rectangle MNPQ with diagonals and See Figure 4.21. PROVE: MP ⬵ NQ NQ MP M Q N P Figure 4.21 PROOF Statements Reasons 1. Rectangle MNPQ with diagonals and 2. MNPQ is a 3. 4. 5. and are right 6. 7. 8. MP ⬵ NQ 䉭NMQ ⬵ 䉭PQM ∠NMQ ⬵ ∠PQM ∠s ∠PQM ∠NMQ MQ ⬵ MQ MN ⬵ QP ⵥ NQ MP 1. Given 2. By definition, a rectangle is a with a right angle 3. Opposite sides of a are 4. Identity 5. By Corollary 4.3.1, the four of a rectangle are right 6. All right are 7. SAS 8. CPCTC ⬵ ∠s ∠s ∠s ⬵ ⵥ ⵥ 쮿 Discover Given a rectangle MNPQ like a sheet of paper, draw diagonals and . From a second sheet, cut out formed by two sides and a diagonal of MNPQ. Can you position so that it coincides with ? ANSWER 䉭 NQP 䉭 MPQ 䉭 MPQ NQ MP Yes Exs. 1–4 THE SQUARE All rectangles are parallelograms; some parallelograms are rectangles; and some rectangles are squares. A B D C Square ABCD Figure 4.22 A square is a rectangle that has two congruent adjacent sides. See Figure 4.22. DEFINITION Because a square is a type of rectangle, it has four right angles and its diagonals are con- gruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular. In Chapter 8, we measure area in “square units.” THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhom- bus is rhombi pronounced rho˘m-bi¯. All sides of a rhombus are congruent. COROLLARY 4.3.4 The diagonals of a rhombus are perpendicular. THEOREM 4.3.5 A rhombus is a parallelogram with two congruent adjacent sides. DEFINITION Exs. 5–7 In Figure 4.23, the adjacent sides and of rhombus ABCD are marked con- gruent. Because a rhombus is a type of parallelogram, it is also necessary that and . Thus, we have Corollary 4.3.4. AD ⬵ BC AB ⬵ DC AD AB We will use Corollary 4.3.4 in the proof of the following theorem. D C A B Figure 4.23 EXAMPLE 2 Study the picture proof of Theorem 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS. PICTURE PROOF OF THEOREM 4.3.5 D A a B E Figure 4.24 D A b B E D A c E GIVEN: Rhombus ABCD, with diagonals and [See Figure 4.24a]. PROVE: PROOF: Fold across to coincide with [see Figure 4.24b]. Now fold across half-diagonal to coincide with [see Figure 4.24c]. The four congruent triangles formed in Figure 4.24c can be unwrapped to return rhombus ABCD of Figure 4.24a. With four congruent right angles at vertex E, we see that . AC ⬜ DB 䉭AED DE 䉭CED 䉭CED AC 䉭ABC AC ⬜ DB DB AC 쮿 Geometry in the Real World The jack used in changing an automobile tire illustrates the shape of a rhombus. Discover Sketch regular hexagon RST VWX. Draw diagonals and . What type of quadrilateral is RT VX? ANSWER XV RT Rectangle