C.P.C.T.C Corresponding Parts of a Congruent Triangles are Congruent <AMB & <BMC are right angles <ABM< ≅ <MBC Corresponding Parts of a Congruent Triangles are Congruent C.P.C.T.C Find a pair of ∆ with parts you are trying to prove are ≅ Prove the two triangles are congruent by C.P.C.T.C corresp. parts are congruent proving a bisector or angle bisector proving lines are parallel or perpendicular prove AB ∥ CD prove the 2 angles it bisects are congruent. B A C B Statements Reasons 1) BE ∥ DF, BE ≅ DF, CE ≅ AF 1) given C E A D F M Statements Reasons 1) <AMB and <BMC are right angles, <ABM ≅ <MBC 1) given

1) <AMB and <BMC are right angles, <ABM ≅ <MBC 1) given prove the 2 angles it bisects are congruent. B A C M Statements Reasons 1) <AMB and <BMC are right angles, <ABM ≅ <MBC 1) given 2) <AMB ≅ <BMC 2) Right angle congruence theorem 3) BM ≅ BM 3) Reflexive Property 4) ∆AMB ≅ ∆CMB 4) A S A 5) AM ≅ MC 5) CPCTC 6) BM is a bisector of AC 7) Definition of a segment bisector

prove AB ∥ CD B V V Statements Reasons C F E D A 1) given 1) BE ∥ DF, BE ≅ DF, CE ≅ AF 2) <BEA ≅ <DFC 2) alternate exterior angles 3) AE + EF = AF CF + EF = CE 3) Betweenness of points 4) AE + EF = CF + EF 4) substitution 5) subtraction 5) AE = CF 6) ∆BEA≅∆DFC 6) SAS 7) <BAE ≅ <DCF 7) <C.P.C.T.C 8) AB ∥ CD 8) If 2 lines form ≅ alt. int. angles, they are ∥

Things that can be proven with C.P.C.T.C What am I trying to prove?? Here are the steps to do it! Show triangles are ≅ Prove segments or angles within the ∆ are ≅ by C.P.C.T.C segments or angles are congruent Show triangles are ≅ Are the parts of bisecting segment or angle contained in the ∆? Prove those parts are ≅ by C.P.C.T.C (this proves the bisector) segments or angles bisector Show triangles are ≅ Are the angles in the ∆ alt. interior, corresponding, or alt. exterior angles of lines? Prove those parts are ≅ by C.P.C.T.C (this proves the bisector) Lines are parallel ∥ Show triangles are ≅ Are the angles contained in ∆ and formed by the lines linear pairs? Prove those angles are ≅ by C.P.C.T.C (this proves you have right angles and lines must be parallel) Lines are perpendicular ⊥

perpendicular bisector Altitude The altitude is a ____________ segment from one ___________ to the other side. perpendicular vertex Every triangle has exactly ______ altitudes. 3 An altitude can be inside OR ___________ of a triangle. altitude outside You may need to extend the bottom in order to draw the altitude. A median is a segment from a ______________ to the ___________ of the other side. Median vertex midpoint median An altitude and median combined into one is a perpendicular bisector Every triangle has exactly ______ medians and all 3 ____________ at a single point called a centroid. 3 intersect

You already know “x” so, substitute it in. Rolling Ball Challenge #1 What do you need to know? You need to find the value of “x” so you can determine what the measurement for angle (7x) is. This will tell you which way the horizontal line will tilt. 2) What do already know? Because we have parallel lines... angles (12x) and (12y +6) are congruent, alternate, interior angles. So... 12x=12y +6 Using algebra to solve for “x” we find x=(y+0.5) 3) Now what? Our parallel lines also tell us angle (8y) and angle(4x+10) are supplementary interior angles on the same side as a transversal. 8y + 4x+10 = 1800 You already know “x” so, substitute it in. 8y + 4(y+0.5) +10 = 180 Remember: x=(y+0.5) 8y+4y+2+10=180 solve for x 12y +12=180 x=(y+0.5) 12y=168 x=(14.5+0.5) y=14.5 x = 15 4) Now, back to the angle that determines tilt... (7x) <7x = 1050 Let’s think about this: but this is what This is what a horizontal our 7x angle would look like. line would look like. Can you see how it would tilt to the left?? 1050 750