Classifying Triangles Measuring Angles in Triangles

1) Name a pair of consecutive interior angles. 2) If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent. 3) If line l is parallel to line AB, name a pair of supplementary angles. 4) If line AB represents the x- axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined. AB l C

1) Name a pair of consecutive interior angles. EX: <5 and <4 <6 and <3 2) If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent. EX: <1 and <2 because they are corresponding angles. < 9 and <3 because they are alternate interior angles. AB l C

3) If line l is parallel to line AB, name a pair of supplementary angles. EX: <9 and <2 <5 and <4 4) If line AB represents the x- axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined. Negative AB l C

Triangle- A three-sided polygon Polygon- A closed figure in a plane that is made up of segments. Acute Triangle- All the angles are acute. Obtuse Triangle- One angle is obtuse. Right Triangle- One angle is right. HypotenuseLeg

Equlangular Triangle- An acute triangle in which all angles are congruent. Scalene Triangle- No two sides are congruent. Isosceles Triangle- At least two sides are congruent. Equilateral Triangle- All the sides are congruent. Angle Sum Theorem- The sum of the measures of the angles of a triangle is 180. Vertex angle Leg Base angle Base

Auxiliary Line- A line or line segment added to a diagram to help in a proof. Third Angle Theorem- If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Exterior Angle- An angle that forms a linear pair with one of the angles of the polygon. Interior Angle- An angle inside a polygon. Exterior angle Interior angles

Remote Interior Angles- The interior angles of the triangle not adjacent to a given exterior angle. Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Exterior angle Remote Interior angles Corollary- A statement that can be easily proven using a theorem. Corollary- The acute angles of a right triangle are complementary. Corollary- There can be at most one right or obtuse angle in a triangle.

Example 1: Triangle PQR is an equilateral triangle. One side measures 2x + 5 and another side measures x Find the length of each side. Since it is an equilateral triangle all the sides are congruent. 2x + 5 = x + 35 x + 5 = 35 x = 5 Plug 5 in for x in to either equation. x Each side of the triangle is 40. (2x + 5) (x + 35) R Q P

Example 2: Triangle PQR is an isosceles triangle. <P is the vertex angle, PR = x + 7, RQ = x – 1, and QP = 3x – 5. Find x, PR, RQ, and QP. Since it is an isosceles triangle and we know that <P is the vertex angle, PR is congruent to PQ. PR = PQ x + 7 = 3x = 2x = 2x 6 = x Plug 6 in for x in the equation for PR. PR = x + 7 PR = PR = 13 = PQ (x + 7) (3x - 5) R Q P (x - 1) Plug 6 in for x in the equation for RQ. RQ = x – 1 RQ = 6 – 1 RQ = 5

Example 3: Given triangle STU with vertices S(2,3), T(4,3), and U(3,-2), use the distance formula to prove triangle STU is isosceles. If it is isosceles two of the sides have the same length. The distance formula is d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) ST d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((2 – 4) 2 + (3 – 3) 2 ) d=√((-2) 2 + (0) 2 ) d=√(4 + 0) d=√(4) d= 2 TU d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((4 – 3) 2 + (3 – -2) 2 ) d=√((4 – 3) 2 + (3 + 2) 2 ) d=√((1) 2 + (5) 2 ) d=√(1 + 25) d=√(26) SU d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((2 – 3) 2 + (3 – -2) 2 ) d=√((2 – 3) 2 + (3 + 2) 2 ) d=√((-1) 2 + (5) 2 ) d=√(1 + 25) d=√(26) Since TU and SU are congruent but ST is not this is an isosceles triangle.

Example 4: Given triangle STU with vertices S(2,6), T(4,-5), and U(-3,0), use the distance formula to prove triangle STU is scalene. If it is scalene none of the sides are congruent. The distance formula is d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) ST d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((2 – 4) 2 + (6 – -5) 2 ) d=√((2 – 4) 2 + (6 + 5) 2 ) d=√((-2) 2 + (11) 2 ) d=√( ) d=√(125) TU d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((4 – -3) 2 + (-5 – 0) 2 ) d=√((4 + 3) 2 + (-5 - 0) 2 ) d=√((7) 2 + (-5) 2 ) d=√( ) d=√(74) SU d=√((x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 ) d=√((2 – -3) 2 + (6 – 0) 2 ) d=√((2 + 3) 2 + (6 - 0) 2 ) d=√((5) 2 + (6) 2 ) d=√( ) d=√(61) Since none of the sides are congruent, this is a scalene triangle.

Example 5: A surveyor has drawn a triangle on a map. One angle measures 42 degrees and another measures 53 degrees. Find the measure of the third angle. 42 x53 According to the angle sum theorem, all the angles in a triangle add up to = x 180 = 95 + x 85 = x So the third angle is 85 degrees.

Example 6: A surveyor has drawn a triangle on a map. One angle measures 41 degrees and another measures 74 degrees. Find the measure of the third angle. 41 x74 According to the angle sum theorem, all the angles in a triangle add up to = x 180 = x 65 = x So the third angle is 65 degrees.

Example 7: Find the measure of each numbered angle in the figure if line l is parallel to line m. l 135 m m<1 <1 and 135 are supplementary. 180 = m< = m<1 m<5 <1 and <5 are alternate interior angles so they are congruent. m<1 = m<5 45 = m<5 m<2 <5, <2, and 60 are supplementary. 180 = m<5 + m< = 45 + m< = 105 m<2 75 = m<2 m<3 All the angles in a triangle add up to = m<1 + m<2 + m<3 180 = m<3 180 = m<3 60 = m<3 m<4 <3 and <4 are supplementary. 180 = m<3 + m<4 180 = 60 + m<4 120 = m<4

Example 8: Find x, y, and m<ABC 80 xy (3x – 22) CB A Find x According to the exterior angle theorem the remote interior angles are add up to the exterior angle. 3x – 22 = 80 + x 2x -22 = 80 2x = 102 x = 51 Find y All the angles in a triangle add up to = 80 + x + y 180 = y 180 = y 49 = y m<ABC Plug 51 in for x in the equation for m<ABC. m<ABC = 3x – 22 m<ABC = 3(51) – 22 m<ABC = 153 – 22 m<ABC = 131