4-3 Angle Relationships in Triangles
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1 4-3 ngle Relationships in Triangles Objectives: G.O.0: Prove theorems about triangles. For the board: You will be able to find the measures of interior and exterior angles of triangles and apply these theorems. ell Work 4.3:. Find the m<, if m< = 30, m< = 70, and m< = 80.. What is the complement of an angle with measure 7? 3. How many lines can be drawn through parallel to P? P nticipatory Set: Experiment: Given a triangle, tear off the 3 corners. rrange them so that they are adjacent angles. What do you notice about the sum of these three angles? They make a straight angle and thus add to equal 80 Instruction: The Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 80. Given: Triangle onclusion: m< + m< + m<3 = 80 3 Open the book to page 3 and read example. Example: Use the diagram drawn from to find the indicated angle measures. a. m<xyz b. m<yxw = = = 78 W c. m<ywx + 8 = = 50 6 X Y 40 Z White oard ctivity: Practice: Find m<jk. m<k = 80 ( ) = 44 m<kj = 44 m<jk = 80 ( ) = 3 L 70 J K
2 Practice: Find the measure of each angle of Δ. 3x + x = 80 5x + 50 = 80 5x = 30 x = 6 m< = (6) + 7 = 69 m< = 3(6) = 78 3x (x + 7) 33 n auxiliary line is a line that is added to a figure to aid in a proof. The Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 80. Given: Triangle w/ Prove: m< + m< + m<3 = draw l through. Parallel Postulate. m<4 + m< + m<5 = 80. Linear Pair Postulate (Triple) 3. m<4 = m< 3. lternate Interior ngles 4. m<5 = m<. lternate Interior ngles 5. m< + m< + m<3 = Substitution Property orollary The acute angles of a right triangle are complementary. Given: Δ is a right triangle Prove: m< + m< = 90. m< + m< + m< = 80. Triangle Sum Theorem. < is a right angle. efn. of a rt. Δ 3. m< = efn. of a rt. < 4. m< + m< = Subt. Prop. Read example on page 33. Example: The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? Hint: This is the same as asking what the complement of the given angle is. Recall: complement = 90 - angle a b. a c. (x 4) (90 a) 90 (x 4) x + 4 (94 x)
3 White oard ctivity: Practice: The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? a b. x c. 48 / = 6.3 (90 x) /5 = 4 3/5 dditional Practice: The measure of one acute angle of a right triangle is four times the measure of the other acute angle. Find the measure of each acute angle. Let x and 4x represent the two angles x + 4x = 90 5x = 90 x = 8 4x = 7 orollary The measure of each angle of an equiangular triangle is 60 Explanation: If a triangle is equilateral then it is equiangular. Let the measure of each angle be x, then x + x + x = 80 or 3x = 80. This results in x = 60 When the sides of a triangle are extended, the three original angles are the interior angles. (<, <, <3) The angles that are adjacent to the interior angles are the exterior angles. (<4) < and < are referred to as the remote interior angles of exterior <4. m< + m< + m<3 = 80 and m<3 + m<4 = 80 therefore m< + m< = m<4. This relationship is called the Exterior ngle Theorem. Exterior ngle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote interior) angles of the triangle. Given: triangle Prove: m< + m< = m<4 Open the book to page 33 and read example 3. Example: a. Given m< = 67 and m< = 64 find m<. m< + m< = m< = 3 b. Given m< = 5 and m< = find m<. m< + m< = m< 5 + m< = 5 m< = 63 c. Given m< = x, m< = 7 and m< = (x ), find the value of x. Then find the measure of the exterior angle. 7 + x = x 83 = x
4 exterior angle = (83) = 55 White oard ctivity: Practice: Find m<. z = 6z 9 z + 9 = 6z 9 00 = 4z z = 5 m< = 6(5) 9 = 4 (z + ) (6z - 9) Exterior ngle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote interior) angles of the triangle. Given: triangle Prove: m< + m< = m<4. Δ w/exterior <4. Given. m< + m< + m<3 = 80. Triangle Sum Theorem 3. <3 and <4 are a linear pair 3. efinition of a Linear Pair 4. m<3 + m<4 = Linear Pair Postulate 5. m< + m< + m<3 = m<3 + m<4 5. Substitution Property 6. m< + m< = m<4 6. Subtraction Property Third ngle Theorem If two angles on one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Given: < <X, < <Y Z Prove: < <Z X Y Open the book to page 34 and read example 4. Example: Find m<k and m<j. 4y = 6y 40 y = 50 y = 5 y = 5 m<k = 4(5 ) = 00 and m<j = 00 F K J (4y ) (6y 50) G H I White oard ctivity: Practice: Find m<p and m<t. x = 4x 3 x = 3 x = 6 x = ±4 m<p = m<t = (6) = 3 (x ) P R T S (4x 3)
5 ssessment: Question student pairs. Independent Practice: Text: pgs prob. 4, 9 3, Explorations: pg. 45 prob. 3. For a Grade: Text: pgs prob. 6, 0,, 34.
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