Type of Triangle Definition Drawing. Name the triangles below, and list the # of congruent sides and angles:

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1 Name: Triangles Test Type of Triangle Definition Drawing Right Obtuse Acute Scalene Isosceles Equilateral Number of congruent angles = Congruent sides are of the congruent angles Name the triangles below, and list the # of congruent sides and angles: 1) 2) 3) 4) 95⁰ congruent sides congruent sides congruent sides congruent sides congruent angles congruent angles congruent angles congruent angles Complete the examples below: 1) 2) 3) 5in x x 4ft y 40⁰ y 50⁰ x y x = x = x = y = y = y = 112

2 Isosceles Triangles Name Practice Problems: Find the values of the variables x = x a = y 35 y = 27 a n d n = d = 96 Hint: An angle bisector of the vertex of an isosceles triangle is also a perpendicular bisector of the base 3. t r s r = s = 4. e e = f = t = g = 125 f g x x 42 (4u 7) (2u + 15) 3x - 13 x = u = x = J c = m e b (c + 10) m J = (3c - 6) O m = m O = e = (12c +48) m B = b = B Classify ΔJOB by its sides and angles 113

3 Triangle Facts Triangles can be classified by their angles as well as their sides: By sides they are either: 1. -no sides the same 2. -two sides the same 3. -all sides the same By angles they are either: 1. -One right angle and two acute angles 2. -All three angles are acute 3. -One obtuse angle and two acute angles Right Triangle Facts: 1. Side opposite(across from) the right angle is called the 2. The other two sides are called the 3. The three sides are related by the formula Equilateral Triangle Facts: 1. All three sides are 2. All three angles are 3. Each of the three angles are equal to Isosceles Triangle Facts: 1. Two sides are 2. The two angles opposite the congruent sides are 3. The two sides that are congruent are called the 4. The other side is called the 5. The two angles that are congruent are opposite the 6. The two angles that are congruent are called the 7. The other angle is called the angle Example 1: Classify the following by sides and angles: 1) 2) 3) 4) 114

4 Triangle Rules Interior Angles The angles on the inside of a triangle Drawing Exterior Angles Adjacent Interior Angles Remote Interior Angles Isosceles Vertex Angle Bisector Rule When the sides of a triangle are extended, the angles that are adjacent (next to) the interior angles Angles on the inside of the triangle which are located next to one another Two angles of a triangle that are not adjacent to the exterior angle The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base Equilateral Triangle Rule If a triangle is equilateral, then it is equiangular Remote Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of the two remote interior angles Triangle Inequality Rule The sum of the lengths of any two sides of a triangle is greater than the length of the third side Opposite Angle Rule If the two sides of a triangle are not congruent, than the larger angle lies opposite the longer side. 115

5 Interior / Exterior Angles in Triangles Name Date Period 1. Find x and y. y x Find the measures of the three interior angles of the triangle. Then solve for x (3x + 10) 3. Find all missing angles of the figure Will the following three lengths make a triangle? If no, justify. a. 4, 7, 11 b. 2, 9, 5 c. 15, 14, 10 d. 6, 8, If three sides of a triangle are 14, 6 and x, what is the range of possible values for x? 6. In triangle RST, R = 50 and S = 60. List the sides of each triangle in order from shortest to largest. 7. Which segment is shortest? Note that the figure is not drawn to scale. D A C B 116

6 For problems 7-15, find x (5x) (7x) X 137 (3x) (7x) (3x) (5x + 50) x x x x

7 Triangles Practice TYPE OF TRIANGLE Right Isosceles DRAWING WITH PARTS LABELLED ITEMS TO MARK OR LABEL Right Angle Legs Hypotenuse Legs Base Vertex Angle Base Angle Question: What happens to an isosceles triangle that has had its vertex angle bisected? (Create a drawing and write the rule) More Practice: 1) 2) b 4ft x a c 120⁰ a = b= c = x= 92⁰ 2x x = 146⁰ 3) List the sides largest to smallest 4) Is this triangle possible? Why or why not? a 65⁰ 47⁰ c 68⁰ 3 21 b

8 Lines of Concurrency Define: 1) Concurrent lines- 2) Point of concurrency- Point of Definition Drawing concurrency Angle Bisector Perpendicular Bisector Median Altitude A bisector that divides an angle of a triangle into two congruent adjacent angles A segment that is perpendicular to a side of a triangle at the midpoint of the side A segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side The perpendicular segment from a vertex of a triangle to the opposite side or to the line that contains the opposite side Identify the following: 1) 2) 3) 4) 119

9 Points of Concurrency Name Given the following pictures and markings, identify each of the following as (a) an angle bisector, (b) a perpendicular bisector, (c) an altitude, or (d) a median. List all that apply Identify each of the following as (a) an angle bisector, (b) a perpendicular bisector, (c) an altitude, or (d) a median

10 Name: Date: Hour: Geometry Triangles Review (Part 1) Sketch-Pair- Share: Sketch as many of the following figures as you can in the given time. Get with a partner and share what you have. Team up in groups of four to finish. The group with all figures sketched gets to share. Groups with additional information for a figure can volunteer to share their sketches as well. Equilateral Triangle Isosceles Triangle Scalene Triangle Right Triangle Hypotenuse Legs (Right Triangle) Legs (Isosceles) Base Vertex Angle Base Angles Interior Angles Exterior Angles Vertex Angle Bisector (Isosceles) Linear Pair Vertical Angles Alternate Interior Angles Adjacent Interior Angle Remote Interior Angles Remote Angle Theorem Triangle Inequality Theorem (What has to be true about side lengths?) 121

11 Name: Date: Hour: Triangles Review Part 2 1) 2) 3) x 0 b 0 a y 0 b 0 c 0 d a 0 c 0 c 0 d 0 a= b= a= b= x= y= c= d= c= d= Solve for the variable: 4) 5) 6) 2x (c + 10) (3c - 6) 3x - 12 (4u-11) (2u +5) (2c +14) 80 7) 8) 9) 20 (2x + 20) x y (3y 22) (3x 8)

12 Word Problems: 1. In ABD, AB DB, m A is 12 less than three times a number, and m D is 13 more than twice the same number. Find m B and then make a conjecture about ABD: 2. Find the range of values for the third side of a triangle if the first two sides are 14 and Are the following lengths possible for the sides of a triangle? a. 4, 5, and 2 b. 5, 8, and 3 c. 9, 14, and 3 d. 12,14, and List the sides of Triangle ABC in order from longest to shortest if the m A = 40 and the m B = In DEF, m D = (2x + 4), m E = (6x 58) and the measure of the exterior angle at F is represented by 5x. Find the value of x and then make a conjecture about DEF. Critical Thinking: 1. Is an equilateral triangle also an isosceles triangle? Why or why not? 2. Can a right triangle also be Isosceles? Scalene? Equilateral? 123

13 Identify the following as: angle bisector; perpendicular bisector; altitude or median. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 124

14 Directions: Rate your understanding of each concept with either a plus (understand well), a check (limited understanding or unsure), or a minus (don t know). Write a formula or description for each. Find the unknown angle of a triangle given other two + - x Identify the hypotenuse and legs of a Right Triangle Identify the legs, vertex, and base angles of an Isosceles Triangle A B A B C C Hypotenuse _AC Legs & Legs & Base angles & Vertex Find the measure of the base angles of an Isosceles Triangle when given the vertex angle 50 Find the measure of the vertex angle when given one base angle 40 Solve Algebra problems using properties of Isosceles Triangles 3x-3 2x+5 Solve Algebra problems using properties of an equilateral triangle 4x+4 Solve for the unknown angles of a triangle including exterior angles c 125 a b d 125

15 Solve Algebra Problems using the Remote Angle Theorem 3x 7x 5x+50 Tell whether 3 given lengths are possible sides of a triangle a) 2,4,10 b)3,8,5 c)10,12,21 Give the range of possible lengths of the third side when given 2 sides AB = 5 BC= 10 Range for AC A Identify special segments of a triangle Use properties of a median to solve for unknown sides and angles Use the properties of an altitude to solve for unknown angles of a triangle x+5 2x 2x+4 3x Given 1 = 4 and 4 = 52⁰ Find 1 =, 2 =, 3 = List sides of a triangle in order from least to greatest given 2 angles 46 x y z

Geometry - Concepts 9-12 Congruent Triangles and Special Segments

Geometry - Concepts 9-12 Congruent Triangles and Special Segments Concept 9 Parallel Lines and Triangles (Section 3.5) ANGLE Classifications Acute: Obtuse: Right: SIDE Classifications Scalene: Isosceles: