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Transcript
Name: ________________________________________ Date: __________________ Block: _________ Triangle Congruence Re-Visit Section G.5: Properties of Triangle Lengths and Angle Measures Triangle Basics: • Know how to classify triangles o By Angles: Acute, Obtuse, Right, Equiangular o By Sides: Scalene, Isosceles, Equilateral • Know the triangle theorems: o Triangle Sum Theorem: the sum of the interior angles of a triangle is 180 o Exterior Angle Theorem: the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles o Third Angles Theorem: if 2 angles of one triangle are congruent to 2 angles of another triangle, then the 3rd angles are also congruent o Theorem to Order Angles: largest angle is across from the largest side. o Theorem to Order Sides: largest side is across from the largest angle. o Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side (must be greater, cannot be equal or less). o Hinge Angle Theorem: if you have 2 pair of congruent sides for 2 triangles, the greatest “hinge” angle is across from the longest 3rd side. o Hinge Angle Converse Theorem: if you have 2 pair of congruent sides for 2 triangles, the longest 3rd side is across from the largest “hinge” angle. • Know special triangles and their properties o Isosceles: Parts: Legs (2 congruent sides), base (3rd side), vertex angle (formed by legs), base angles (angles across from legs) Base Angles Theorem: isosceles triangles have congruent base angles Converse of Base Angles Theorem: if 2 angles are congruent, the triangle is isosceles o Equilateral: All 3 sides are congruent Corollary to the Base Angles Theorem: equilateral triangles are equiangular Corollary to the Converse of the Base Angles Theorem: equiangular triangles are equilateral • Know triangle segment/centers vocabulary/properties o Midsegment: parallel to third side, half the length of third side o Perpendicular bisector: intersects a triangle side perpendicularly at its midpoint; center is the circumcenter o Angle bisector: creates two congruent angles at a vertex of the triangle; center the incenter o Median: intersects the vertex of a triangle and the midpoint of the opposite side; center is centroid; distance from centroid to vertex is 2/3 of entire segment length o Altitude: called the height of the triangle; center is orthocenter Name: ________________________________________ Date: __________________ Block: _________ Triangle Congruence Re Re-Visit Example Problems: 1. Can you form a triangle with the given lengths? If no, use the triangle inequality to explain why not. a) 2, 6, 19 b) 3, 5, 8 2. Find all possible lengths of the 3rd side of the triangle with given length lengths (be sure to include units): a) 11ft, 23 ft b) 10m, 6m 3. List the sides from smallest to largest: 4. List the angles from smallest to largest: 5. Find the value of x. Then classify the triangle by its angles: 6. Complete the statement with <, >, or =. MN _____ LK 7. Find the value of x: 8. DE is a midsegment of ABC. Find x and y. 9. Find x: 10. Q is the centroid of XYZ. a) Find XQ: b) Find XN: c) Find XM: Name: ________________________________________ Date: __________________ Block: _________ Triangle Congruence Re Re-Visit Section G.6: Triangle Congruence Proving Triangles are Congruent: o Triangles can be proven congruent by: SSS post postulate, SAS postulate ulate, HL theorem (special case of SSA that proves congruency) congruency), ASA postulate, AAS theore eorem. o Triangles cannot be proven congruent by: AAA or SSA o Once triangles are re proven congruent, corresponding parts can be concluded congruent by Corresponding Parts of Congruent ongruent Triangles are Congruent ongruent (CPCTC). (CPCTC) o Third Angles Theorem: if 2 angles of one triangle are congruent to 2 angles of another triangle, then the 3rd angles are also congruent o Base Angles Theorem and Converse Converse:: Two sides of a triangle are congruent IFF the angles opposite them are congruent. o Corollaries to Base Angle Theorem and Converse Converse:: A triangle is equilateral IFF it is equiangular. Example problems: 1. State the missing congruence statement to prove ∆ABD ≅ ∆CDB CDB by the stated theorem/ postulate. Given: ∆ABC is isosceles. ∠ A and ∠ C are base angles. a) Prove by: ASA. b) Prove by: HL 3. Are the triangles congruent? If so, what congruence postulate / theorem can be used? If not, why not? 2. ABC ≅ VTU. Find m<T. Find AB. 4. Are the triangles congruent? If so, what congruence postulate / theorem can be used? If not, why not? 5. Use the coordinates to determine if ABC LMN. If not, why not? A (-5, 1), B (-4, 4), C (-2, 2, 3), L (0, 2), M (1, 5), N (3, 4) [Do not graph! Hint: Use the distance formula on all sides] 6. Write a proof: 7. Write a proof: Given: ≅ ≅ Given: ABC is isosceles; Prove: ABD ≅ Prove: ≅ CBD bisects <ABC.