A wall that is 12 ft long by 8 ft high has a triangular brace Find the perimeter of the square shown.

In Figure 4.34, , , and are altitudes of trapezoid HJKL. The length of any altitude of HJKL is called the height of the trapezoid. KQ JP , LY HX Exs. 1–6 P H L Q J X Y K Figure 4.34 Discover Using construction paper, cut out two trapezoids that are copies of each other. To accomplish this, hold two pieces of paper together and cut once left and once right. Take the second trapezoid and turn it so that a pair of congruent legs coincide. What type of quadrilateral has been formed? Parallelogram ANSWER The preceding activity may provide insight for a number of theorems involving the trapezoid. The base angles of an isosceles trapezoid are congruent. THEOREM 4.4.1 EXAMPLE 2 Study the picture proof of Theorem 4.4.1. PICTURE PROOF OF THEOREM 4.4.1 GIVEN: Trapezoid RSTV with and [See Figure 4.35a]. PROVE: and PROOF: By drawing and we see that Theorem 4.1.6. By HL, so CPCTC. in Figure 4.35a because these angles are supplementary to congruent angles and . ∠T ∠V ∠R ⬵ ∠S ∠V ⬵ ∠T 䉭RYV ⬵ 䉭SZT RY ⬵ SZ SZ ⬜ VT , RY ⬜ VT ∠R ⬵ ∠S ∠V ⬵ ∠T RS ‘ VT RV ⬵ ST T V R S a Figure 4.35 T V Y Z R S b Some of the glass panels and trim pieces of the light fixture are isosceles trapezoids. Other glass panels are pentagons. 쮿 Geometry in the Real World The following statement is a corollary of Theorem 4.4.1. Its proof is left to the student. The diagonals of an isosceles trapezoid are congruent. COROLLARY 4.4.2 The length of the median of a trapezoid equals one-half the sum of the lengths of the two bases. THEOREM 4.4.3 If diagonals and were shown in Figure 4.36 at the left, they would be congruent. BD AC EXAMPLE 3 Given isosceles trapezoid ABCD with see Figure 4.36: a Find the measures of the angles of ABCD if and . b Find the length of each diagonal not shown if it is known that and . Solution a Because , , so and . Then or , and or . Subtracting , we determine the supplements of A and B. That is, . b By Corollary 4.4.2, , so . Then and . Thus, . Also . BD = 19 - 8 = 11 AC = 28 - 5 = 11 y = 8 3y = 24 2y - 5 = 19 - y AC ⬵ BD m ∠C = m∠D = 54° ∠s 180 - 126 = 54 126° m ∠B = 108 + 46 126° m ∠A = 128 + 30 x = 8 2x = 16 12x + 30 = 10x + 46 m ∠A = m∠B BD = 19 - y AC = 2y - 5 m ∠B = 10x + 46 m ∠A = 12x + 30 AB ‘ DC 쮿 A B C D Figure 4.36 For completeness, we state two properties of the isosceles trapezoid.

1. An isosceles trapezoid has line symmetry; the axis of symmetry is the

perpendicular-bisector of either base.

2. An isosceles trapezoid is cyclic; the center of the circle containing all four

vertices of the trapezoid is the point of intersection of the perpendicular bisectors of any two consecutive sides or of the two legs. The proof of the following theorem is left as Exercise 33. We apply Theorem 4.4.3 in Examples 4 and 5. NOTE: The length of the median of a trapezoid is the “average” of the lengths of the bases. Where m is the length of the median and and are the lengths of the bases, . m = 1 2 b 1 + b 2 b 2 b 1 EXAMPLE 4 In trapezoid RSTV in Figure 4.37, and M and N are the midpoints of and , respectively. Find the length of median if and . VT = 18 RS = 12 MN TS RV RS ‘ VT The proof of Theorem 4.4.4 is left as Exercise 28. In Figure 4.37, and Theorems 4.4.5 and 4.4.6 enable us to show that a quadrilateral with certain char- acteristics is an isosceles trapezoid. We state these theorems as follows: MN ‘ VT. MN ‘ RS The median of a trapezoid is parallel to each base. THEOREM 4.4.4 If two base angles of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. THEOREM 4.4.5 If the diagonals of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. THEOREM 4.4.6 Solution Using Theorem 4.4.3, , so , or . Thus, . MN = 15 MN = 1 2 30 MN = 1 2 12 + 18 MN = 1 2 RS + VT 쮿 EXAMPLE 5 In trapezoid RSTV, and M and N are the midpoints of and , respectively see Figure 4.37. Find MN, RS, and VT if , , and . Solution Using Theorem 4.4.3, we have , so Then and . Now , so . Also, ; therefore, . Finally, ; therefore, . NOTE: As a check, leads to the true statement . 25 = 1 2 20 + 30 MN = 1 2 RS + VT VT = 210 + 10 = 30 VT = 2x + 10 MN = 25 MN = 3x - 5 = 310 - 5 RS = 20 RS = 2x = 210 x = 10 3x - 5 = 2x + 5 3x - 5 = 1 2 [2x + 2x + 10] or 3x - 5 = 1 2 4x + 10 MN = 1 2 RS + VT VT = 2x + 10 MN = 3x - 5 RS = 2x TS RV RS ‘ VT 쮿 Exs. 7–12 R S T N M V Figure 4.37 1 R S T X V Figure 4.38 Consider the following plan for proving Theorem 4.4.5. See Figure 4.38. GIVEN: Trapezoid RSTV with and PROVE: RSTV is an isosceles trapezoid PLAN: Draw auxiliary line parallel to . Now show that , so in . But in parallelogram RXTS , so and RSTV is isosceles. RV ⬵ ST RX ⬵ ST 䉭RXV RV ⬵ RX ∠V ⬵ ∠1 ST RX ∠V ⬵ ∠T RS ‘ VT Theorem 4.4.6 has a lengthy proof, for which we have provided a sketch. GIVEN: Trapezoid ABCD with and [See Figure 4.39a on page 208.] PROVE: ABCD is an isosceles trapezoid. PLAN: Draw and in Figure 4.39b. Now we can show that ABEF is a rectangle. Because by HL. Then 䉭AFC ⬵ 䉭BED AF ⬵ BE, BE ⬜ DC AF ⬜ DC AC ⬵ DB AB ‘ DC A B C D a Figure 4.39 A B C E F D b by CPCTC. With by Identity, by SAS. Now because these are corresponding parts of and . Then trapezoid ABCD is isosceles. For several reasons, our final theorem is a challenge to prove. Looking at parallel lines a, b, and c in Figure 4.40, one sees trapezoids such as ABED and BCFE. However, the proof whose “plan” we provide uses auxiliary lines, parallelo- grams, and congruent triangles. 䉭BDC 䉭ACD AD ⬵ BC 䉭ACD ⬵ 䉭BDC DC ⬵ DC ∠ACD ⬵ ∠BDC Exs. 13–15 A B C a t m b c F E D S R Figure 4.40 If three or more parallel lines intercept congruent line segments on one transversal, then they intercept congruent line segments on any transversal. THEOREM 4.4.7 GIVEN: Parallel lines a, b, and c cut by transversal t so that ; also transversal m in Figure 4.40 PROVE: PLAN: Through D and E, draw and . In each formed, and Given it follows that By AAS, we can show ; then by CPCTC. DE ⬵ EF 䉭DER ⬵ 䉭EFS DR ⬵ ES. AB ⬵ BC, ES ⬵ BC. DR ⬵ AB ⵥ ES ‘ AB DR ‘ AB DE ⬵ EF AB ⬵ BC EXAMPLE 6 In Figure 4.40, . If and , find EF. Solution Using Theorem 4.4.7, we find that . 쮿 EF = 8.4 DE = 8.4 AB = BC = 7.2 a ‘ b ‘ c Exs. 16, 17 Exercises 4.4

1. Find the measures of the remaining angles of trapezoid

ABCD not shown if and and .

2. Find the measures of the remaining angles of trapezoid

ABCD not shown if and and .

3. If the diagonals of a trapezoid are congruent, what can

you conclude about the trapezoid?

4. If two of the base angles of a trapezoid are congruent,

what type of trapezoid is it? m ∠D = 118° m ∠B = 63° AB ‘ DC m ∠C = 125° m ∠A = 58° AB ‘ DC

5. What type of quadrilateral is formed when the midpoints

of the sides of an isosceles trapezoid are joined in order?

6. In trapezoid ABCD, is the median. Without writing a

formal proof, explain why . MN = 1 2 AB + DC MN A B C N M X W Y Z D