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Chapter 5 Applying Congruent Triangles Warm Up For Chapter 5 5.2 Right Triangles Internet Activity 5.1 Special Segments in Triangles Objective: Identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle How will I use this? Special segments are used in triangles to solve problems involving engineering, sports and physics. Chapter 5 Median Perpendicular Bisector Altitude An example to tie it all together Click Me!! Angle Bisector A segment that connects a vertex of a triangle to the midpoint of the side opposite the vertex. A line segment with 1 endpoint at a vertex of a triangle and the other on the line opposite that vertex so that the line segment is perpendicular to the side of the triangle. Perpendicular Bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side. Theorems Theorem 5.1: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Theorem 5.2: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. Chapter 5 Median Perpendicular Bisector Altitude Angle Bisector Theorems Theorem 5.3: Any point on the bisector of an angle is equidistant from the sides of the angle. Theorem 5.4: Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. Chapter 5 Median Perpendicular Bisector Altitude Angle Bisector Warm UP In ABC , m A 2 x 5 , m B 3 x 15 , and m C 5 x 10 Find the value of x and the measure of each angle. Warm Up Answers x 20 m A 45 m B 45 m C 90 How did I get that? Click the answer to see! BONUS!!! What type of triangle is ABC? Click me to find the Answer!! Section 5.1 BONUS!!! What type of triangle is ABC? Click me to find the Answer!! Because the question give you angle measures, we take the sum of the angles and set them equal to 180. } } } 2 x 5 3 x 15 5 x 10 180 mB mA mC 10 x 20 180 Combine like terms! 10 x 200 Add 20 to both sides! x 20 Divide by 10 on both sides! Chapter 5 mA mB mC Section 5.1 Use substitution for the answer you found for x and plug it into the equation for angle A. m A 2x 5 x 20 m A 2 ( 20 ) 5 40 5 45 Chapter 5 mB mC Section 5.1 Use substitution for the answer you found for x and plug it into the equation for angle B. m B 3 x 15 x 20 m B 3 ( 20 ) 15 60 15 45 Chapter 5 mA mC Section 5.1 Use substitution for the answer you found for x and plug it into the equation for angle C. m C 5 x 10 x 20 m C 5 ( 20 ) 10 100 10 90 Chapter 5 mA mB Section 5.1 Triangle ABC is a right isosceles triangle Why is that?? Chapter 5 Section 5.1 Angle Bisector What is an Angle Bisector? Click me to find out! Move my vertices around and see what happens!! Angle Bisector Theorems Example Section 5.1 Median Example Draw the three medians of triangle ABC. Name each of them. B Answer A C Median Example Draw the three medians of triangle ABC. Name each of them. B F A D E C Back to Section 5.1 Altitude Example Draw the three altitudes, QU, SV, and RT. R Answer Q S Altitude Example Draw the three altitudes, QU, SV, and RT. V U Q R T S Back to Section 5.1 Perpendicular Bisector Example Draw the three lines that are perpendicular bisectors of XYZ. Y Answer X Label the lines l, Z m, and n. Perpendicular Bisector Example Draw the three lines that are perpendicular bisectors of XYZ. Y m X l n Z Back to Section 5.1 Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. B 3x 2 A x 6 2x 3 D 3x 2 C Answer Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. B 3x 2 x 4 x 6 AC 21 Show me how you got those answers! A 2x 3 D 3x 2 C Back to Section 5.1 Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. Means that the angle is split into 2 congruent parts. Set the two angles equal to each other and solve. A B 3x 2 x 6 2x 3 D Once you find x, plug it into AD and DC. Since you are looking for the total length, AC, use segment addition to find the total length. 3x 2 C Back to Section 5.1 5.1 Proofs Together YOU TRY!!! Given: RAB is isosceles with vertex angle RAB EA is the bisector of RAB Prove: EA is a median 2 1 RAB is isosceles RA BA 1 5 3 EA is the bisector of RAB RAE RAE BAE BAE 6 4 RE BE AE AE 7 1. Given 2. Def of Isos Triangle 3. Def of Angle Bisector 4. Reflexive 5. SAS 6. CPCTC 7. Def of Median AE is a median 5.1 Proofs Given: STU is equilatera l TW is an angle bisector of STU Prove: TW is a median of STU 2 1 STU is equilatera l TU TS 1 5 UTW STW 3 TW is angle bisector of STU UTW STW 6 4 UW SW TW TW 7 5. SAS 1. Given 2. Def of Equilateral Triangle 6. CPCTC 7. Def of Median 3. Def of Angle Bisector 4. Reflexive T W is a median We’re done, take me back to the beginning! Example SGB has vertices S(4,7) G(6,2) and B(12,-1) Determine Keep clicking to see graph! the coordinate s of point J on GB so that median of SGB SJ is the median Midpoint See the Work!! S G B SGB has vertices S(4,7), G(6,2) and B(12,-1) What is the Midpoint Formula? Midpoint of GB 6 12 2 1 , 2 2 1 9, 2 x 2 x1 y 2 y 1 , 2 2 Next Question Point M has coordinate Is GM an altitude s (8,3). of SGB ? GM must be to SB Slope of GM Slope of SB 32 86 1 7 12 4 1 2 What can we conclude? 1 We’re done, take me back to the beginning! 5.2 Right Triangles An Internet Activity CLICK TO BEGIN Take notes as you read along with each Theorem or Postulate!! Leg Leg Theorem Click on the triangle and learn about the Theorems or Postulates. Hypotenuse Angle Theorem Leg Angle Theorem Hypotenuse Leg Postulate Click me when done Examples I finished! Click me!! Solving for variables Stating additional information Solve for… Example 1 Example 3 Example 2 State the additional information. Example Example Example 1 2 3 Find the value of x and y so that DEF PQR by HA. P D 2x 3 F 3 y 10 4x 1 E Q 4 y 20 Answer R 3 y 10 4 y 20 y 30 2x 3 4x 1 2 x 2 x 1 Find the value of x and y so that DEF PQR by LL. E 56 23 F D 2x 4 Q 2y 9 R P Answer 23 2 y 9 56 2 x 4 32 2 y 52 2 x 16 y 26 x Find the value of x and y so that DEF PQR by LA. y 3 E FP 2x 4 6 D R 3 y 20 Q Answer 3 y 20 y 50 2 y 30 y 15 2x 4 6 2 x 10 x5 Back to Beginning State the additional information needed to prove the pair of triangles congruent by LA. J M K L Answer Proving triangles congruent by LA means a leg and an angle of the right triangle must be congruent. J M LM LK M K K L OR JL JL MJL KJL Next Example State the additional information needed to prove the pair of triangles congruent by HA. S V T X Z Y Answer State the additional information needed to prove the pair of triangles congruent by HA. S V T X Z Y The keyword was additional. When proving triangles congruent by HA, all that is needed is to show that the hypotenuse is congruent on each triangle as well as an acute angle. In these triangles both are already shown so there is no ADDITIONAL information needed. Next Example State the additional information needed to prove the pair of triangles congruent by LA. B A C D F Answer State the additional information needed to prove the pair of triangles congruent by LA. B C C F BC DF A D OR BAC FAD BA AD F Back to Beginning