Given: bisects Given: E

Each triangle has three altitudes—one from each vertex. As these are shown for in Figure 3.26, the three altitudes seem to meet at a common point. We now consider the proof of a statement that involves the corresponding altitudes of congruent triangles; corresponding altitudes are those drawn to corresponding sides of the triangles. 䉭ABC A B C Figure 3.26 A B D C Figure 3.27 R H S T Figure 3.25 R S V T D a F E Figure 3.28 M b P N R c T S Corresponding altitudes of congruent triangles are congruent. THEOREM 3.3.1 PROOF Statements Reasons 1. 1. Given Altitudes to and to 2. and 2. An altitude of a 䉭 is the line segment from one vertex drawn to the opposite side 3. and are right 3. If two lines are , they form right 4. 4. All right angles are 5. and 5. CPCTC from 6. 6. AAS 7. 7. CPCTC CD ⬵ TV 䉭CDA ⬵ 䉭TVR 䉭ABC ⬵ 䉭RST ∠A ⬵ ∠R AC ⬵ RT ⬵ ∠CDA ⬵ ∠TVR ∠s ⬜ ∠s ∠TVR ∠CDA ⬜ TV ⬜ RS CD ⬜ AB RS TV AB CD 䉭ABC ⬵ 䉭RST GIVEN: Altitudes to and to See Figure 3.27. PROVE: CD ⬵ TV RS TV AB CD 䉭ABC ⬵ 䉭RST Each triangle has three medians—one from each vertex to the midpoint of the opposite side. As the medians are drawn for in Figure 3.28a, it appears that the three medians intersect at a point. 䉭DEF Each triangle has three angle bisectors—one for each of the three angles. As these are shown for in Figure 3.28b, it appears that the three angle bisectors have a point in common. See Figure 3.28 on page 146. Each triangle has three perpendicular bisectors for its sides; these are shown for in Figure 3.28c. Like the altitudes, medians, and angle bisectors, the perpendi- cular bisectors of the sides also meet at a single point. The angle bisectors like the medians of a triangle always meet in the interior of the triangle. However, the altitudes like the perpendicular bisectors of the sides can meet in the exterior of the triangle; see Figure 3.28c. These points of intersection will be given greater attention in Chapter 7. The Discover activity at the left opens the doors to further discoveries. In Figure 3.29, the bisector of the vertex angle of isosceles is a line segment of symmetry for . 䉭ABC 䉭ABC 䉭RST 䉭MNP Exs. 1–6 EXAMPLE 1 Give a formal proof of Theorem 3.3.2. The bisector of the vertex angle of an isosceles triangle separates the triangle into two congruent triangles. THEOREM 3.3.2 GIVEN: Isosceles , with bisects See Figure 3.29. PROVE: 䉭ABD ⬵ 䉭CBD ∠ABC BD AB ⬵ BC 䉭ABC PROOF Statements Reasons 1. Isosceles with 1. Given 2. bisects 2. Given 3. 3. The bisector of an separates it into two 4. 4. Identity 5. 5. SAS 䉭ABD ⬵ 䉭CBD BD ⬵ BD ⬵ ∠s ∠ ∠1 ⬵ ∠2 ∠ABC BD AB ⬵ BC 䉭ABC Recall from Section 2.4 that an auxiliary figure must be determined. Consider Figure 3.30 and the following three descriptions, which are coded D for determined, U for underdetermined, and O for overdetermined: D: Draw a line segment from A perpendicular to so that the terminal point is on . [Determined because the line from A perpendicular to is unique; see Figure 3.30a.] U: Draw a line segment from A to so that the terminal point is on . [Underdetermined because many line segments are possible; see Figure 3.30b.] O: Draw a line segment from A perpendicular to so that it bisects . [Overdetermined because the line segment from A drawn perpendicular to will not contain the midpoint M of ; see Figure 3.30c.] BC BC BC BC BC BC BC BC BC Discover Using a sheet of construction paper, cut out an isosceles triangle. Now use your compass to bisect the vertex angle. Fold along the angle bisector to form two smaller triangles. How are the smaller triangles related? ANSWER They are congruent. A C D B 1 2 Figure 3.29 B A C a Figure 3.30 B A C b B A C M ? ? c 쮿 In Example 2, an auxiliary segment is needed. As you study the proof, note the uniqueness of the segment and its justification reason 2 in the proof. General Rule: An early statement of the proof establishes the “helping line” as the altitude or the angle bisector or whatever else. Illustration: See the second line in the proof of Example 2. The chosen angle bisector leads to congruent triangles, which enable us to complete the proof. STRATEGY FOR PROOF 왘 Using an Auxiliary Line EXAMPLE 2 Give a formal proof of Theorem 3.3.3. If two sides of a triangle are congruent, then the angles opposite these sides are also congruent. THEOREM 3.3.3 GIVEN: Isosceles with [See Figure 3.31a.] PROVE: NOTE: Figure 3.31b shows the auxiliary segment. ∠M ⬵ ∠N MP ⬵ NP 䉭MNP PROOF Statements Reasons 1. Isosceles with 1. Given 2. Draw bisector from P to 2. Every angle has one and only one bisector 3. 3. The bisector of the vertex angle of an isosceles 䉭 separates it into two 4. 4. CPCTC ∠M ⬵ ∠N ⬵ 䉭s 䉭MPQ ⬵ 䉭NPQ MN PQ ∠ MP ⬵ NP 䉭MNP P M N a Figure 3.31 P M N b Q Theorem 3.3.3 is sometimes stated, “The base angles of an isosceles triangle are congruent.” We apply this theorem in Example 3. EXAMPLE 3 Find the size of each angle of the isosceles triangle shown in Figure 3.32 on page 149 if: a b The measure of each base angle is 5° less than twice the measure of the vertex angle m ∠1 = 36° 쮿 In some instances, a carpenter may want to get a quick, accurate measurement without having to go get his or her tools. Suppose that the carpenter’s square shown in Figure 3.33 is handy but that a miter box is not nearby. If two marks are made at lengths of 4 inches from the corner of the square and these are then joined, what size angle is determined? You should see that each angle indicated by an arc measures 45°. Example 4 shows us that the converse of the theorem “The base angles of an isosce- les are congruent” is also true. However, see the accompanying Warning. 䉭 Solution a . Since and and are , we have Now , and . b Let the vertex angle measure be given by x. Then the size of each base angle is . Because the sum of the measures is 180°, Therefore, and . 쮿 m ∠2 = m∠3 = 71° m ∠1 = 38° 2x - 5 = 238 - 5 = 76 - 5 = 71 x = 38 5x = 190 5x - 10 = 180 x + 2x - 5 + 2x - 5 = 180 2x - 5 m ∠2 = m∠3 = 72° m ∠1 = 36° m ∠2 = 72 2m ∠2 = 144 36 + 2m ∠2 = 180 ⬵ ∠3 ∠2 m ∠1 = 36° m ∠1 + m∠2 + m∠3 = 180° 1 2 3 Figure 3.32 ? ? Figure 3.33 V a T U Figure 3.34 Discover Using construction paper and scissors, cut out an isosceles triangle MNP with . Fold it so that coincides with . What can you conclude? ANSWER ∠N ∠M MP ⬵ PN ∠M ⬵∠ N Warning The converse of an “If, then” statement is not always true. EXAMPLE 4 Study the picture proof of Theorem 3.3.4. If two angles of a triangle are congruent, then the sides opposite these angles are also congruent. THEOREM 3.3.4 PICTURE PROOF OF THEOREM 3.3.4 GIVEN: with [See Figure 3.34a.] PROVE: PROOF: Drawing [see Figure 3.34b], we see that by AAS. Now by CPCTC. VU ⬵ VT 䉭VPT ⬵ 䉭VPU VP ⬜ TU VU ⬵ VT ∠T ⬵ ∠U 䉭TUV T U V b P When all three sides of a triangle are congruent, the triangle is equilateral. If all three angles are congruent, then the triangle is equiangular. Theorems 3.3.3 and 3.3.4 can be used to prove that the sets {equilateral triangles} and {equiangular triangles} are equivalent. Exs. 7–17 쮿 Many of the properties of triangles that were investigated in earlier sections are summarized in Table 3.1. An equilateral triangle is also equiangular. COROLLARY 3.3.5 An equiangular triangle is also equilateral. COROLLARY 3.3.6 The perimeter of a triangle is the sum of the lengths of its sides. Thus, if a, b, and c are the lengths of the three sides, then the perimeter P is given by . See Figure 3.36. P = a + b + c DEFINITION An equilateral or equiangular triangle has line symmetry with respect to each of the three axes shown in Figure 3.35. Geometry in the Real World X Y Z Figure 3.35 a c b Figure 3.36 Braces that create triangles are used to provide stability for a bookcase. The triangle is called a rigid figure. EXAMPLE 5 GIVEN: and FIND: The perimeter of Solution If , then . Therefore, P = 14.2 P = 3.6 + 5.3 + 5.3 P = a + b + c AC = AB = 5.3 ∠B ⬵ ∠C 䉭ABC BC = 3.6 AB = 5.3 ∠B ⬵ ∠C Exs. 18–22 A B C a c b Figure 3.37 쮿 TABLE 3.1 Selected Properties of Triangles Equilateral Scalene Isosceles equiangular Acute Right Obtuse Sides No two are Exactly two All three Possibly Possibly Possibly are are two or three two sides; two sides sides Angles Sum of is Sum of is Sum of All acute; One right ; One obtuse 180° 180°; two is 180°; three sum of sum of ; sum of is 180°; is 180°; is 180°; possibly two possibly possibly or three two two ; acute acute are complementary ∠s ∠s ∠s ⬵ ∠s ⬵ ⬵ 45° ∠s ⬵ 60° ⬵ ∠s ∠s ∠ ∠s ∠s ∠ ∠s ∠s ∠s ∠s c 2 = a 2 + b 2 ⬵ ⬵ ⬵ ⬵ ⬵ ⬵ Exercises 3.3 For Exercises 1 to 8, use the accompanying drawing.

1. If

what type of triangle is ?

2. If

which angles of are congruent?

3. If

, which sides of are congruent?

4. If , ,

and , what is the perimeter of ?

5. If and ,

find .

6. If and ,

find .

7. If and ,

find .

8. If and

, find . In Exercises 9 to 12, determine whether the sets have a subset relationship. Are the two sets disjoint or equivalent? Do the sets intersect?

9. L = {equilateral triangles}; E = {equiangular triangles} 10. S = {triangles with two

⬵ sides}; A = {triangles with two } 11. R = {right triangles}; O = {obtuse triangles} 12. I = {isosceles triangles}; R = {right triangles} ⬵ ∠s m ∠T m ∠V = 40° VU ⬵ VT m ∠V m ∠T = 72° VU ⬵ VT m ∠V m ∠T = 69° VU ⬵ VT m ∠U m ∠T = 69° VU ⬵ VT 䉭VTU TU = 8 VU = 10 VU ⬵ VT 䉭VTU ∠T ⬵ ∠U 䉭VTU VU ⬵ VT, 䉭VTU VU ⬵ VT, In Exercises 13 to 18, describe the segment as determined, underdetermined, or overdetermined. Use the accompanying drawing for reference. V T U Exercises 1–8 A B m Exercises 13–18

13. Draw a segment through point A. 14. Draw a segment with endpoints A and B.

15. Draw a segment parallel to line m.

16. Draw a segment perpendicular to m.

17. Draw a segment from A perpendicular to m. 18. Draw

so that line m bisects .

19. A surveyor knows that a lot has the shape of an isosceles

triangle. If the vertex angle measures 70° and each equal side is 160 ft long, what measure does each of the base angles have? AB AB AB AB

20. In concave quadrilateral ABCD, the angle at A measures

40°. is isosceles, bisects , and bisects . What are the measures of , , and ? ∠1 ∠ADC ∠ABC ∠ADB DC ∠ABD BC 䉭ABD A B D C 1 3 2 1

31. Suppose that . Also,

bisects and bisects . Are the corresponding angle bisectors of congruent triangles congruent? ∠FDE DY ∠CAB AX 䉭ABC ⬵ 䉭DEF In Exercises 21 to 26, use arithmetic or algebra as needed to find the measures indicated. Note the use of dashes on equal sides of the given isosceles triangles.

21. Find and if .

m ∠3 = 68° m ∠2 m ∠1

22. If , find

, the angle formed by the bisectors of and .

23. Find the measure of , which is formed by the

bisectors of and . Again let .

24. Find an expression for the

measure of if and the segments shown bisect the angles of the isosceles triangle.

25. In isosceles with vertex A

not shown, each base angle is 12° larger than the vertex angle. Find the measure of each angle.

26. In isosceles not shown,

vertex angle A is 5° more than one-half of base angle B. Find the size of each angle of the triangle. In Exercises 27 to 30, suppose that is the base of isosceles not shown.

27. Find the perimeter of if

and .

28. Find AB if the perimeter of is 36.4 and

.

29. Find x if the perimeter of is 40,

, and .

30. Find x if the perimeter of is 68,

, and . BC = 1.4x AB = x 䉭ABC BC = x + 4 AB = x 䉭ABC BC = 14.6 䉭ABC BC = 10 AB = 8 䉭ABC 䉭ABC BC 䉭ABC 䉭ABC m ∠3 = 2x ∠5 m ∠3 = 68° ∠3 ∠1 ∠5 ∠2 ∠3 m ∠4 m ∠3 = 68°

3 4

2 5 1 Exercises 22–24 C X B A Exercises 31, 32 F Y E D

32. Suppose that ,

is the median from A to , and is the median from D to . Are the corresponding medians of congruent triangles congruent? In Exercises 33 and 34, complete each proof using the drawing below.

33. Given:

Prove: AB ⬵ AC ∠3 ⬵ ∠1 EF DY BC AX 䉭ABC ⬵ 䉭DEF D B E C A 2 6

3 1

7 H Exercises 33, 34 PROOF Statements Reasons 1. 1. ? 2. ? 2. If two lines intersect, the vertical formed are 3. ? 3. Transitive Property of Congruence 4. 4. ? AB ⬵ AC ⬵ ∠s ∠3 ⬵ ∠1

34. Given:

Prove: ∠6 ⬵ ∠7 AB ⬵ AC PROOF Statements Reasons 1. ? 1. Given 2. 2. ? 3. and are 3. ? supplementary; and are supplementary 4. ? 4. If two are supplementary to , they are to each other ⬵ ⬵ ∠s ∠s ∠7 ∠1 ∠6 ∠2 ∠2 ⬵ ∠1 In Exercises 35 to 37, complete each proof.

35. Given:

Prove: is isosceles HINT: First show that .

36. Given:

M is the midpoint of and Prove:

37. Given: Isosceles

with vertex P Isosceles with vertex Q Prove:

38. In isosceles triangle BAT, . Also,

. If and