Prove Theorems about Lines and Angles
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1 GEOMETRY Prove Theores about Lines and Angles OJECTIVE #: G.CO.9 OJECTIVE Prove theores about lines and angles. Theores include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segent are exactly those equidistant fro the segent s endpoints. IG IDEA (Why is this included in the curriculu?) To establish angle and length relationships as they relate to parallel and perpendicular lines. PREVIOUS KNOWLEDGE (What skills do they need to have to succeed?) The student ust understand parallel lines. The student ust understand the types of transforations. VOCAULARY USED IN THIS OJECTIVE (What ters will be essential to understand?) PREVIOUS VOCAULARY (Ters used but defined earlier) Angle: A geoetric figure fored by rotating a ray about its initial point. Corresponding Angles: Two angles that are fored by two lines and a transversal which lie in the sae position with respects to the lines and the transversal. Equidistant: Two points that are the sae distance fro a given point. Parallel Lines: Two lines that are coplanar and do not intersect. Perpendicular: Two lines/segents/rays that intersect to for right angles. Perpendicular isector: A perpendicular line/segent/ray that intersects a segent at its idpoint. NEW VOCAULARY (New Ters and definitions introduced in this objective) Alternate Interior Angles: Two angles that are fored by two lines and a transversal and that lie between the two lines on opposite sides of the transversal. Alternate Exterior Angles: Two angles that are fored by two lines and a transversal and that lie outside the two lines on opposite sides of the transversal.
2 Copleentary Angles: Two angles whose easures have the su 90. [G.CO.10] Consecutive Interior Angles (Sae Side Interior Angles): Two angles that are fored by two lines and a transversal and that lie between the two lines on the sae side of the transversal. Linear Pair: Two adjacent angles whose non-coon sides are opposite rays, thus aking the suppleentary. [G.CO.10] Suppleentary Angles: Two angles whose easures have the su 180. [G.CO.10] Transversal Line: A line that intersects two or ore coplanar lines at different points Vertical Angles: The two non-adjacent angles fored by intersecting lines. SKILLS (What will they be able to do after this objective?) Students will be able to identify all of the types of angle relationships fored when two parallel lines are cut by a transversal. Students will be able to know all of the relationships between pairs of angles. Students will be able to prove that all points on a perpendicular bisector of a segent are equidistant fro the segent endpoints. SHORT NOTES (A short suary of notes so that a teacher can get the basics of what is expected.) This objective is introduced in geoetry using the transforations described in G.CO.4 by translating along vectors.
3 A o For exaple, corresponding angles are fored by translating one angle along the transversal onto its iage. See attached files for suarization that ay be used for this objective. o AXY is translated along XY X Y A o This results in A and A being parallel lines, therefore creating congruent corresponding angles AXY and A X Y. X X A Y o Students should be encouraged to explore the different types of transforations or coposite transforations that for these pairs of angles. Have the students construct the perpendicular bisector of a segent. Allow students to choose points along the bisector and easure the distance fro the endpoints of the segent. o Exaple: Student drew C and constructed the perpendicular bisector. Student then chose point A and easured the distance fro A to C and fro A to. Students can copare their findings (that AC A ) to deterine this holds true for any point along the perpendicular bisector. elow is the foral two-colun proof. Given: AC with perpendicular bisector D Prove: A C STATEMENT D is the bisector of AD CD AD & CD are right s AD CD D D REASON Given Definition of a isector (D is the idpoint of AC ) Definition of a isector ( D is to AC ) All Right Angles are Reflexive Property (Coon Side) A D C
4 AD CD A C SAS CPCTC (Corresponding Parts of Congruent Triangles are Congruent) MISCONCEPTIONS (What are the typical errors or difficult areas? Also suggest ways to teach the.) Although these pairs of angles are fored without the lines being parallel, the congruence and suppleentary aspects are not applicable if the original lines are not parallel. FUTURE CONNECTIONS (What will they use these skills for later?) These angle relationships of parallel lines and perpendicular bisector will be utilized to discover the characteristics of different parallelogras. ADDITIONAL EXTENSIONS OR EXPLANATIONS (What needs greater explanation?) Students will utilize congruent triangles to prove all points on a perpendicular bisector of a segent are equidistant fro the segent endpoints. Corresponding Angles: Two angles fored by translating a given angle in a plane. ASSESSMENTS (Questions that get to the heart of the objective ultiple choice, short answer, ulti-step) 1) Correctly atch the following and include the transforation that aps one angle onto its pair. 1. Alternate Interior Angles E A. 4 and 6 Translate 3 along the transversal onto 7. Then rotate 180 onto 6 2. Alternate Exterior Angles C. 1 and 5 Rotate and then translate along the transversal onto 7 3. Corresponding Angles C. 2 and 7 Translate 1 along the transversal onto 5 4. Consecutive Interior Angles A Not a transforation D. 1 and 4 5. Vertical Angles D E. 3 and 6 Rotate onto 4 2) Find the easure of angle a and b if t. Justify your reasoning using transforations. l b 125 t a a = 125 by translating the given angle along the transversal b = 125 by rotating the given angle 180
5 3) Find the easure of angle a and b if t. Justify your reasoning using transforations. 65 b t a The given angle and b for a linear pair, therefore b = 115. Rotate b 180 and translate along the transversal onto a. Therefore a = 115 Fro CCSD Geoetry Honors Seester 1 Practice Exa Use the diagra. s r n Which stateent would be used to prove lines r and s are parallel? (A) 1 and 3 are congruent () 2 and 7 are copleentary (C) 4 and 1 are congruent (D) 8 and 6 are suppleentary 2. In the diagra, n and p q. p q n x 88 What is the value of x? (A) 44 () 88 (C) 92 (D) 176
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