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10 Oct 2016 8:00 - 9:30 AM Geometry Proposed Agenda 1) Bulletin clarkmagnet.net 2) Angles of a Triangle Homework corrections from Oct 4 - Doc Camera 3) Homework from Oct 6 - discuss and decide next step 4) Review pages 17 and 18 information 5) Isosceles Triangles (includes a new construction - have yellow booklet and compass and straightedge ready) 6) Homework Homework: Find a Picture Find a real world picture of congruent triangles on the internet and print it out, or draw a picture from life - or take your own picture and print that out. Explain how you know the triangles are congruent. Postulates Segment addition postulate: If B is between A and C, then AB + BC = AC A B C Angle Addition Postulate: If P is in the interior of angle RST, then m∠RST = m ∠RSP +m∠PST R P S T If angle RST is a straight angle and B is any point not on the line is in the interior of angle RST, then m∠RSB + m ∠BST = 180° B R S T Through any 2 points there is exactly one line What’s missing on my diagram? Through any 2 points there is exactly one line Letters for the the points and arrows to show this is line, not a line segment Read the bold headings so you have an idea what information you can find on these 2 pages when you need it. For today’s work - highlight Definition of Isosceles triangle Isosceles Triangles A triangle with AT LEAST two congruent sides. G-CO Congruent angles in isosceles triangles Below is an isosceles triangle ABC with A |AB| = |AC| B C Three students propose different arguments for why m∠(B) = m∠(C) Could they all be correct? A B C Ravi: If I draw the bisector of ∠A then this is a line of symmetry for triangle ABC and so m∠B = m∠C. Is he correct? A B C Reflection about the line will map A onto itself; line segment AB onto AC and AC onto AB - the angle at the vertex has been cut in half and AB and AC are the same length as each other. A Reflections are rigid transformations so length is not changed. Now we see we have also interchanged the angles at B and C, so they must be congruent which means they have equal measures. B C Brittney: If M is the midpoint of BC triangle ABM is congruent to triangle ACB and so ∠B and ∠C are congruent. Is she correct? A B M C M is the midpoint of line segment BC which means BM = MC using the Midpoint theorem (or definition of midpoint). A B M C M is the midpoint of line segment BC which means BM = MC using the Midpoint theorem (or definition of midpoint). We know by the reflexive property that AM = AM. A B M C By Side-Side-Side Congruence postulate we can see that △ABM is congruent to △ACB. A B M C By Side-Side-Side Congruence postulate we can see that △ABM is congruent to △ACM. A Since ∠B and ∠C are corresponding parts of congruent triangles they are congruent. B M C Courtney: If P is a point on BC such that line AP is perpendicular to BC then triangle ABP is congruent to triangle ACP and so ∠B = ∠C. Is she correct? A B P C Theorem: If two lines are perpendicular, they form congruent adjacent angles. We know AP = AP by reflexive property. By Pythagorean theorem we know that BP = PC SSS Congruence Theorem, triangle ABP is congruent to triangle ACP. Since ∠B and ∠C are corresponding parts of congruent triangles they are congruent. A B P C Are there any other ways to show this? A B C Isosceles Triangle Definition? Given AB = AC Prove: ∠ABC = ∠ACB Draw line segment AD to bisect angle BAC. Statement Reason 1. AB = AC 1. Given 2. ∠BAD = ∠CAD Statement Reason 1. AB = AC 1. Given 2. ∠BAD = ∠CAD 2. Definition of angle bisector 3. AD = AD Statement Reason 1. AB = AC 1. Given 2. ∠BAD = ∠CAD 2. Definition of angle bisector 3. AD = AD 4. △BAD =△CAD 3.Reflexive property Statement Reason 1. AB = AC 1. Given 2. ∠BAD = ∠CAD 2. Definition of angle bisector 3. AD = AD 3.Reflexive property 4. △BAD =△CAD 4. SAS congruence postulate 5. ∠ABC = ∠ACB Statement Reason 1. AB = AC 1. Given 2. ∠BAD = ∠CAD 2. Definition of angle bisector 3. AD = AD 3.Reflexive property 4. △BAD =△CAD 4. SAS congruence postulate 5. ∠ABC = ∠ACB 5. Corresponding parts of congruent triangles are congruent. Isosceles Triangle - bottom of page Theorem. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Draw it: A perpendicular bisector Is a ray, line, or line segment that bisects and is perpendicular to a line segment. DC _|_ AB D AC = CB Both have to be true. C Construct A perpendicular bisector? What would we do? (Try in yellow booklet) 1) 2) Perpendicular bisector Given AB = AC, ∠BAD = ∠CAD Prove: BC = CD, BC _|_ AD 2 Column proof, complete, then start homework page 23. Statement Reason 1. AB = AC, ∠BAD =∠CAD 1. Given 2. AD = AD 2. Reflexive property 3. △ABD = △ACD 3. SAS congruence postulate 4 BD = CD 4. Corresponding parts of congruent triangles are congruent (CPCTC) 5. ∠ADB = ∠ADC 5. CPCTC Statement 1. AB = AC, ∠BAD =∠CAD Reason 1. Given 2. AD = AD 2. Reflexive property 3. △ABD = △ACD 3. SAS congruence postulate 4 BD = CD 4. Corresponding parts of congruent triangles are congruent (CPCTC) 5. ∠ADB = ∠ADC 5. CPCTC 6. BC _|_ AD 6. If two lines form congruent adjacent angles then the lines are perpendicular. Homework Complete Properties of Isosceles Triangles Homework problems 1 - 3 on page 23. Show your thinking clearly and neatly. Bring you congruent triangles in the real world photographs or drawings on Wednesday for use on a class poster, if you did not turn yours in today in class.