Slide 1 / 183 Slide 2 / 183 Geometry Congruent Triangles 2015-10-23 www.njctl.org Slide 3 / 183 Slide 4 / 183 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. click on the topic to go · Congruent Triangles to that section MP1: Making sense of problems & persevere in solving them. · Proving Congruence MP2: Reason abstractly & quantitatively. · SSS Congruence MP3: Construct viable arguments and critique the reasoning of · SAS Congruence others. MP4: Model with mathematics. · ASA Congruence MP5: Use appropriate tools strategically. · AAS Congruence MP6: Attend to precision. · HL Congruence MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. · Triangle Congruence Proofs · CPCTC Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on · Isosceles Triangle Theorem this slide) with a reference to the standards used. · PARCC Sample Questions If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 4 (Answer) / 183 Slide 5 / 183 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Congruent MP3: Construct viable arguments and critique the reasoning of Math Practice others. Triangles MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math [This object is a pull tab] Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. Return to Table of Contents If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
Slide 6 / 183 Slide 7 / 183 Similar Triangles Congruent Triangles We learned in the Similar Triangles topic (Triangles unit) that if Congruent triangles are a special case of similar triangles. two triangles are similar: The constant of proportionality is one, so the corresponding · All their angles are congruent sides are of equal measure. · All their corresponding sides are in proportion For congruent triangles, all the angles are congruent AND all We also learned how to identify the corresponding sides as the corresponding sides are congruent. being opposite to equal angles, or subtended by equal angles And we learned that the constant of proportionality for the corresponding sides of one triangle to the other was called "k." If needed, go back to review that topic before proceeding. Slide 8 / 183 Slide 8 (Answer) / 183 Naming Congruent Triangles Naming Congruent Triangles Just as in the case of similar triangles, the naming of Just as in the case of similar triangles, the naming of congruent triangles is important: order matters. congruent triangles is important: order matters. MP6 Math Practice The statement: The statement: ΔABC is congruent to ΔDEF ΔABC is congruent to ΔDEF Make sure that the students understand that the order in which you name the indicates that these triangles are congruent. indicates that these triangles are congruent. triangles matters. This slide explains why. AND that these angle measures are equal: AND that these angle measures are equal: m ∠ A = m ∠ D m ∠ A = m ∠ D m ∠ B = m ∠ E m ∠ B = m ∠ E [This object is a pull tab] m ∠ C = m ∠ F m ∠ C = m ∠ F AND these lengths are equal: AND these lengths are equal: AB = DE AB = DE BC = EF BC = EF CA = FD CA = FD Slide 9 / 183 Slide 10 / 183 Third Angle Theorem Proving Triangles Congruent Recall the proof showing if we know that two pairs of corresponding angles are congruent, then the third pair of corresponding angles are congruent as well. We can prove triangles congruent by proving the measures of all three corresponding angles and the lengths of all three Statement Reason corresponding sides are equal. 1 ∠ A ≅ ∠ D and ∠ B ≅ ∠ E Given Earlier we showed that we need to prove only two angles are congruent to show that triangles are similar, since the third 2 m ∠ A = m ∠ D; m ∠ B = m ∠ E Definition of ≅ angles angle must then be congruent. 3 m ∠ A+ m ∠ B + m ∠ C = 180º Triangle Sum Theorem There are similar shortcuts to proving triangles congruent. m ∠ D+ m ∠ E + m ∠ F = 180º 4 m ∠ D+ m ∠ E + m ∠ C = 180º Substitution Property of Equality m ∠ D+ m ∠ E + m ∠ F = 180º 5 m ∠ D + m ∠ E + m ∠ C = Substitution Property of Equality m ∠ D + m ∠ E + m ∠ F Subtraction Property of 6 m ∠ C = m ∠ F Equality
Slide 11 / 183 Slide 12 / 183 Corresponding Parts Corresponding Parts Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts Let's review identifying the corresponding parts A D (angles and sides) of pairs of triangles. C F B E Slide 12 (Answer) / 183 Slide 13 / 183 Corresponding Parts Corresponding Parts Given that ΔABC is congruent to ΔDEF, the triangles are Given that ΔABC is congruent to ΔDEF, identify all the marked accordingly in this diagram. congruent corresponding parts A D A D AB ≅ DE ∠ A ≅ ∠ D BC ≅ EF C F Answer ∠ B ≅ ∠ E C F CA ≅ FD ∠ C ≅ ∠ F This example addresses MP6. [This object is a pull tab] B E B E Slide 14 / 183 Slide 15 / 183 Example A D ΔABC ≅ ΔEDC Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and C sides. (Draw a diagram) B Corresponding Sides Corresponding Angles E Part Corresponding Part Segment AB Segment ED ∠ A ∠ E Segment AC Segment EC ∠ B ∠ D Segment CB Segment CD ∠ ACB ∠ ECD
Slide 15 (Answer) / 183 Slide 16 / 183 Example 1 What is the corresponding part to ∠ J ? A ∠ R Have students arrive at the answers as a class, or independently. Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and B ∠ K sides. (Draw a diagram) Corresponding Angles Corresponding Sides C ∠ Q ~ AB LM = ~ BC MN D = ∠ P ~ Corresponding Angles Teacher Notes Corresponding Sides AC LN = J P This example addresses MP5 & MP6 Additional Q's that address MP standards: How could you start this problem? (MP1) How could you use a drawing to assist with this problem? (MP5) How can you make sure that your answer is accurate? (MP6) [This object is a pull tab] K L R Q ΔJKL ≅ ΔPQR Slide 16 (Answer) / 183 Slide 17 / 183 1 What is the corresponding part to ∠ J ? 2 What is the corresponding part to ∠ Q? A ∠ R A ∠ R B ∠ K B ∠ K C ∠ Q C ∠ Q D ∠ P Answer D D ∠ P J P J P [This object is a pull tab] K L R Q K L R Q ΔJKL ≅ ΔPQR ΔJKL ≅ ΔPQR Slide 17 (Answer) / 183 Slide 18 / 183 2 What is the corresponding part to ∠ Q? 3 What is the corresponding part to QP? A JL A ∠ R B LK B ∠ K C KJ C ∠ Q Answer D PQ D ∠ P B J P J P [This object is a pull tab] K L R Q ΔJKL ≅ ΔPQR K L R Q ΔJKL ≅ ΔPQR
Slide 18 (Answer) / 183 Slide 19 / 183 4 The congruence statement for the two triangles is: 3 What is the corresponding part to QP? A JL A ΔBVC ≅ ΔXCZ B LK B ΔXCB ≅ ΔBCX C KJ C ΔVBC ≅ ΔZXC Answer D PQ D ΔCBV ≅ ΔCZX C J P B X [This object is a pull tab] V Z K L R Q C ΔJKL ≅ ΔPQR Slide 19 (Answer) / 183 Slide 20 / 183 5 Complete the congruence statement: ΔXYZ ≅ 4 The congruence statement for the two triangles is: A ΔXWZ A ΔBVC ≅ ΔXCZ B ΔZWX B ΔXCB ≅ ΔBCX C ΔWXZ D ΔZXW C ΔVBC ≅ ΔZXC Answer D ΔCBV ≅ ΔCZX C W Z B X [This object is a pull tab] Z V Y C X Slide 20 (Answer) / 183 Slide 21 / 183 5 Complete the congruence statement: ΔXYZ ≅ Properties of Congruence and Equality A ΔXWZ B ΔZWX We will be using the three properties of congruence we learned earlier C ΔWXZ D ΔZXW Reflexive Property of Congruence Symmetric Property of Congruence Answer Transitive Property of Congruence B W Z As well as the four properties of equality we learned earlier Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality [This object is a pull tab] Substitution Property of Equality Y X
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