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Transcript
4.1: Apply Triangle Sum Properties Aim: To classify triangles and find measures of their angles. What is a triangle? Classifying Triangles by Sides Scalene Triangle: – Has no congruent sides. abc Isosceles Triangle: - Has at least two congruent sides. ca Equilateral Triangle: - Has three congruent sides. Classifying Triangles by Angles Acute Triangle: - Has three acute angles. Right Triangle: - Has one right angle. Obtuse Triangle: - Has one obtuse angle. Equiangular Triangle: - Has three congruent angles. Graph Triangle ABC Graph A(-5, 4), B(2, 6), and C(4, -1). Classify the triangle by its sides using the distance formula. d x2 x1 y2 y1 2 AB 53 BC 53 AC 106 This is an Isosceles 2 d A • x2 x1 2 y2 y1 2 B • • C How can we check if any of the angles are right angles? Using the coordinates A(-5, 4), B(2, 6), and C(4, -1). Find the slope of each line. 2 AB 7 7 BC 2 5 AC 9 What type of triangle is this considered to be? Right Isosceles Triangle Mini-Activity Move you. Draw your desk to the person next to 1 straight line (use your ID card for a straight edge). Take the Triangle give to you and tear off each angle. Theorems Triangle Sum Theorem: – The sum of the measures of the interior angles of a triangle is 180°. Exterior Angle Theorem: – The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Corollary to the Triangle Sum Theorem: – The acute angles of a right triangle are complementary. 4.2: Apply Congruence and Triangles Date: 11/2 Aim: To identify congruent triangles. Do Now:Find m JKM. Find an angle measure Find m JKM. STEP 1 Write and solve an equation to find the value of x. (2x – 5)° = 70° + x° x = 75 Apply the Exterior Angle Theorem. Solve for x. STEP 2 Substitute 75 for x in 2x – 5 to find m JKM. 2x – 5 = 2 75 – 5 = 145 ANSWER The measure of JKM is 145°. Find angle measures from a verbal description Use the corollary to set up and solve an equation. x° + 2x° = 90° x = 30 Corollary to the Triangle Sum Theorem Solve for x. ANSWER So, the measures of the acute angles are 30° and 2(30°) = 60° . 3. Find the measure of 1 in the diagram shown. ANSWER The measure of 1 in the diagram is 65°. Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. 4.2: Apply Congruence and Triangles Date: 11/8/10 Aim: To identify congruent triangles. Do Now: Take out Homework. 4.3: Prove Triangles Congruent by SSS Date: 11/9/10 Aim: To use the side lengths to prove triangles are congruent. Postulate Side-Side-Side (SSS) Congruence Postulate –If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. 4.4: Prove Triangles Congruent by SAS and HL Date: 11/10/10 Aim: To use sides and angle lengths to prove congruence. Do Now: Side-Angle-Side Congruence Postulate (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. 4.4: Prove Triangles Congruent by SAS and HL Date: 11/12/10 Aim: To use sides and angle lengths to prove congruence. Hypotenuse-Leg Congruence Theorem (HL) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. 4.5: Prove Triangles Congruent by ASA and AAS Date: 11/15/10 Aim: To use two more methods to prove congruence (ASA and AAS). Angle-Side-Angle Congruence Postulate (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side Congruence Theorem (AAS) If two angles and a non-included side of one triangle are congruent to the two angles and the corresponding non-included side of a second triangle, then the two triangle are congruent. 4.6: Use Congruent Triangles Date: 12/1/09 Aim: To use congruent triangles to prove corresponding parts congruent. 4.6: Use Congruent Triangles Date: 12/2/09 Aim: To use congruent triangles to prove corresponding parts congruent. 4.7: Use Isosceles and Equilateral Triangles Date: 12/3/09 Aim: To use theorems about isosceles and equilateral triangles. Base Angles Theorem If two sides of a triangle are congruent, then the angels opposite them are congruent. Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. Corollaries Corollary to the Base Angle Theorem: – If a triangle is equilateral, then it is equiangular. Corollary to the Converse of the Base Angles Theorem: – If a triangle is equiangular, then it is equilateral.