4.3 Isosceles and Equilateral

Transcription

1 4.3 Isosceles and quilateral Triangles Goal Use properties of isosceles and equilateral triangles. Key Words legs of an isosceles triangle base of an isosceles triangle base angles Geo-ctivity Properties of Isosceles Triangles 1 Fold a sheet of paper in half. 2 Unfold and label the angles Use a straightedge to draw a as shown. Use a protractor line from the fold to the bottom to measure ah and ak. edge. ut along the line to form What do you notice? an isosceles triangle. J H K 3 Repeat Steps 1 and 2 for different isosceles triangles. What can you say about ah and ak in the different triangles? Stud t H lp VOULRY TIP Isos- means equal, and -sceles means leg. So, isosceles means equal legs. The Geo-ctivity shows that two angles of an isosceles triangle are always congruent. These angles are opposite the congruent sides. The congruent sides of an isosceles triangle are called legs. The other side is called the base. The two angles at the base of the triangle are called the. base angles leg base angles base leg Isosceles Triangle THORM 4.3 ase ngles Theorem Words If two sides of a triangle are congruent, then the angles opposite them are congruent. Symbols If &* c &*, then a c a. 4.3 Isosceles and quilateral Triangles 185

2 XMPL 1 Use the ase ngles Theorem Find the measure of al. Solution ngle L is a base angle of an isosceles triangle. From the ase ngles Theorem, al and an have the same measure. NSWR The measure of al is 52. L M? 52 N Rock and Roll Hall of Fame, leveland, Ohio THORM 4.4 onverse of the ase ngles Theorem Words If two angles of a triangle are congruent, then the sides opposite them are congruent. Symbols If a ca, then &* c &*. Visualize It! ase angles don t have to be on the bottom of an isosceles triangle. XMPL 2 Use the onverse of the ase ngles Theorem Find the value of x. Solution y the onverse of the ase ngles Theorem, the legs have the same length. F onverse of the ase ngles Theorem Substitute 3 for and 12 for F. Subtract 3 from each side. 12 F 3 NSWR The value of x is 9. Use Isosceles Triangle Theorems Find the value of y y y 9 y hapter 4 Triangle Relationships

3 Student Help LOOK K For the definition of equilateral triangle, see p THORMS4.5 and quilateral Theorem Words If a triangle is equilateral, then it is equiangular. Symbols If &* c &* c &*, then a ca ca. 4.6 quiangular Theorem Words If a triangle is equiangular, then it is equilateral. Symbols If a ca ca, then &* c &* c &*. onstructing an quilateral Triangle You can construct an equilateral triangle using a straightedge and compass. 1 raw &*. raw an arc with center that passes through. 2 raw an arc with center that passes through. 3 The intersection of the arcs is point. T is equilateral y the Triangle Sum Theorem, the measures of the three congruent angles in an equilateral triangle must add up to 180. So, each angle in an equilateral triangle measures 60. XMPL 3 Find the Side Length of an quiangular Triangle Find the length of each side of the equiangular triangle. Solution The angle marks show that TQRT is equiangular. So, TQRT is also equilateral Sides of an equilateral T are congruent. Subtract 2x from each side. 3(10) 30 Substitute 10 for x. NSWR ach side of TQRT is 30. 3x R P 2 10 T 4.3 Isosceles and quilateral Triangles 187

4 4.3 xercises Guided Practice Vocabulary heck 1. What is the difference between equilateral and equiangular? Skill heck Tell which sides and angles of the triangle are congruent. 2. M 3. R 4. U L N T S W V Find the value of x. Tell what theorem(s) you used cm x cm x 50 Practice and pplications xtra Practice See p Finding Measures Find the value of x. Tell what theorem(s) you used H J G F Homework Help xample 1: xs. 7 9, 14, 15, 17 19, 27, 28 xample 2: xs xample 3: xs Using lgebra Find the value of x x (5 7) 15. 3x x 4x 188 hapter 4 Triangle Relationships

5 You be the Judge 16. Someone in your class tells you that all equilateral triangles are isosceles triangles. o you agree? Use theorems or definitions to support your answer. IStudent Help I LSSZON.OM HOMWORK HLP xtra help with problem solving in xs is at classzone.com Using lgebra Find the measure of a Using lgebra Find the value of y y y 5 y 4y 3 2y y y 5 5y 14 8y 10 4y 2 Sports 26. hallenge In the diagram at the right, TXYZ is equilateral and the following pairs of segments are parallel: XY &* and LK&*; ZY&* and LJ&; XZ &* and JK&. escribe a plan for showing that TJKL must be equilateral. J Y K X L Z ROK LIMING The climber is using a method of rock climbing called top roping. If the climber slips, the anchors catch the fall. pplication Links LSSZON.OM Rock limbing In one type of rock climbing, climbers tie themselves to a rope that is supported by anchors. The diagram shows a red and a blue anchor in a horizontal slit in a rock face. 27. If the red anchor is longer than the blue anchor, are the base angles congruent? 28. If a climber adjusts the anchors so they are the same length, do you think that the base angles will be congruent? Why or why not? 4.3 Isosceles and quilateral Triangles 189

6 Z X Y Tiles In xercises 29 31, use the diagram at the left. In the diagram, VX &** c WX &** c YX&* c ZX&*. 29. opy the diagram. Use what you know about side lengths to mark your diagram. W V 30. xplain why axwv caxvw. 31. Name four isosceles triangles. 32. Technology Use geometry software to complete the steps. 1 onstruct circle. 2 raw points and on the circle. 3 onnect the points to form T. Is T isosceles? Measure the sides of the triangle to check your answer. Standardized Test Practice Multiple hoice In xercises 33 and 34, use the diagram below. 33. What is the measure of af? What is the measure of af? F 50 G 70 H 125 J F G Mixed Review ngle isectors &( is the angle bisector. Find ma and ma. (Lesson 2.2) Vertical ngles Find the value of the variable. (Lesson 2.4) ( 8) 55 ( 20) lgebra Skills valuating Square Roots valuate. (Skills Review, p. 668) (2 1) hapter 4 Triangle Relationships