Isosceles, Equilateral, Right Triangles

I. Isosceles Triangle (at least two sides are congruent) vertex  Legs: the two congruent sides of the triangle. Base: the remaining side of the triangle. leg leg Base Angles: the angles at the base. base ’s Vertex Angle: the angle opposite the base. base

Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite them are congruent. A If AB  AC, then B  C B C

Ex. 1 Find the measure of F and G. m F = 65o base angles thm 65o H

Converse of Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. A If B  C , then AB  AC B C

Ex. 2 Find the value of x. 4x + 4 = 32 converse of base angles thm Q 32 S

ex. 3. The vertex of an isosceles triangle is 60o ex. 3 The vertex of an isosceles triangle is 60o. What is the measure of the base angles. A mB = mC = x Base angles thm 60o x + x + 60 = 180 ∆ sum thm 2x = 180 - 60 2x = 120 x = _120_ 2 x x C B x = 60 mB = mC = 60o

Ex. 4 Find the mB. A 68o xo C B

II. Equilateral Triangle (all sides are congruent) All equilateral triangles are equiangular and all equiangular triangles are equilateral. All 3 sides are congruent and all 3 angles are congruent.

Ex. 3 Find the length of each side of the equiangular triangle. 6x + 3 = 7x – 1 equiangular equilateral triangle 6x – 7x = - 1 – 3 -x = -4 x= 4 Determine the length of the sides by substituting x = 4 into the equation. 7(4) – 1 =28 – 1 = 27 units

Open your Textbooks Pg 188 #8-14 (even) Pg 189 #18-24 (even)

Pythagorean Theorem and the Distance Formula

Hypotenuse (opposite the right angle) Side c a Side b Pythagorean Theorem is written as when a and b represent the lengths of the sides and c represents the length of the hypotenuse. c2 = a2 + b2

Pythagorean Theorem (hypotenuse)2 = (side)2 + (side)2 c2 = a2 + b2

c2 = a2 + b2 c2 = 82 + 62 c2 = 64 + 36 c2 = 100 c = 100 c = 10 cm Ex. 1 Use Pythagorean Theorem to find the missing side. 8 cm c2 = a2 + b2 c2 = 82 + 62 6 cm c c2 = 64 + 36 c2 = 100 c = 100 c = 10 cm

Ex. 2 Find the unknown side. c2 = a2 + b2 132 = x2 + 32 169 = x2 + 9 169 – 9 = x2 160 = x2 x= 160 x 13 cm 3 cm

Ex. 3. Find the length of the hypotenuse of the right triangle Ex. 3 Find the length of the hypotenuse of the right triangle. Do the lengths of the sides of the triangle form a Pythagorean triple?   c2 = a2 + b2 x2 = 202 + 212 x x2 = 400 + 441 21cm x2 = 841 x = 841 x = 29 cm 20cm 20, 21, and 29 form a Pythagorean triple, because they are integers that satisfy c2 = a2 + b2

Open your Textbooks Pg. 195 #2, 4, 10, 14 Pg 197 #35, 36

Converse of Pythagorean Theorem

If c2 < a2 + b2, then ΔABC is an acute triangle. We can use the Converse of the Pythagorean theorem to determine whether a triangle is an acute, obtuse, and right triangle. If c2 < a2 + b2, then ΔABC is an acute triangle. If c2 = a2 + b2, then ΔABC is a right triangle. If c2 > a2 + b2, then ΔABC is an obtuse triangle. *Note: c represents the largest side measure. a c b

∆WXY is an acute triangle Ex1. Classify the triangles as acute, obtuse, or right. Do the Pythagorean Theorem c2 = a2 + b2 112 = 72 + 92 121 = 49 + 81 121 < 130 (121 is less than 130) ∆WXY is an acute triangle X 7cm 11cm 9cm W Y

∆ABC is a right triangle Ex 2. Classify the triangles as acute, obtuse, or right. c2 = a2 + b2 372 = 122 + 352 1369 = 144 + 1225 1369 = 1369 ∆ABC is a right triangle A 12 37 c 35 B

Ex 3. Classify the triangles as acute, obtuse, or right. c2 = a2 + b2 352 = 122 + 302 1225 = 144 + 900 1225 > 1044 ∆WXY is an obtuse triangle 12cm 35cm 30cm

Distance Formula: the distance d between two points A(x1, y1) and B(x2, y2) is Ex. 4 Find the distance between T(5, 2) and R(-4, -1) to the nearest tenth (one decimal place). d = √90

Class work Pg. 195 #5 Pg. 197 #28 Pg. 203 #2, 6, 14, Pg 204 # 18-20, 25