2. Chapter 9 Summary Project By Ernest Lee 12/17/03 Geometry Honors Period 2 Parallelism Triangles Quadrilaterals

3. skew lines-lines that do not lie in the same plane (AB, DE) parallel lines-lines that are coplanar and never intersect (AB, CD) transversal-a line that intersects 2 coplanar lines at 2 different points alternate interior angles-given lines L 1 and L 2 cut by transversal T at points P and Q, let A be a point on L 1 and let B be a point on L 2 so that A and B are on opposite sides of T,  APQ and  PQB are alternate interior angles L1L1 L2L2 T B A P Q A B CD E FG

4. corresponding angles-given 2 lines cut by a transversal, if  x and  y are alternate interior angles, and  y and  z are vertical angles, then  x and  z are corresponding angles y z x interior angles on the same side of the transversal-given 2 lines cut by a transversal, if  x and  y are alternate interior angles,  v and  w are alternate interior angles, and  v and  x form a linear pair, then  x and  w are interior angles on the same side of the transversal x yw v intercept-if a transversal T intersects 2 lines L 1 and L 2 at points A and B, then L 1 and L 2 intercepts segment AB on the transversal L1L1 L2L2 A B T

5. The AIP Theorem Given 2 lines cut by a transversal. If a pair of alternate interior angles are congruent, then the lines are parallel. L1L1 L2L2 T b a P Q In the diagram, given lines L 1 and L 2 cut by the transversal T. If  a   b, then L 1  L 2. The CAP Theorem Given 2 lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel. L1L1 L2L2 T y xP Q In the diagram, given lines L 1 and L 2 cut by the transversal T. If  x   y, then L 1  L 2.

6. The PCA Corollary If 2 parallel lines are cut by a transversal, each pair of corresponding angles are congruent. 56 L1L1 L2L2 T 21 34 78 In the figure, given L 1  L 2 and they are cut by the transversal T. Then  1  5,  2  6,  3  7, and  4  8. The PAI Theorem If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. L1L1 L2L2 T 2 1 P Q 3 4 In the figure, given L 1  L 2 and they are cut by the transversal T. Then  1  2 and  3  4.

7. The AIP Theorem Given: AD bisects  CAB and CA=CD Prove: CD  AB SR 1. AD bisects  CAB, CA=CD 2.  x  y 3.  y  z 4.  x  z 5. CD  AB 1. Given 2. ITT 3. def. of bisector 4. TPE 5. AIP y z x B A CD

8. The CAP Theorem Given: AC=BC and  DCE  B Prove: CE  AB E BA C D SR 1. Given 2. ITT 3. TPE 4. CAP 1. AC=BC,  DCE  B 2.  A   B 3.  A   DCE 4. CE  AB

9. The PAI Theorem Given: CD  AB and E is the midpoint of CB Prove: AB=CD A C B D E 1. CD  AB, E is the midpoint of CB SR 2. EC=EB 3.  CED   BEA 4.  B   C 5. ∆EAB  ∆EDC 1. Given 2. def. of midpoint 3. VAT 4. PAI 5. ASA 6. AB=DC6. CPCTC

10. The PCA Corollary Given: RT=RS and PQ  RS Prove: PQ=PT SR 2.  T  S 3.  PQT   S 4.  T  PQT 1. RT=RS, PQ  RS 5. PQ=PT 1. Given 2. ITT 3. PCA 4. TPE 5. ITT Converse P SR Q T

11. concurrent-2 or more lines are concurrent if there is a single point that is on all of them point of concurrency-the point shared by the concurrent lines is the point of concurrency

12. The 30-60-90 Triangle Theorem If an acute angle of a right triangle is 30 , then the side opposite the angles is half as long as the hypotenuse. In the figure, given ∆ABC with a right angle at C and m  A is 30 , then BC= 1/2 AB. C 60  30  A B The Median Concurrence Theorem The medians of every triangle are concurrent. Their point of concurrency is 2/3rds of the way along each median, from the vertex to the opposite side. C EB F A D P In the figure, given ∆ABC, let D be the midpoint of CB, E be the midpoint of AB, and F be the midpoint of AC. Then the medians AD, BF, and CE intersect at P so that AP= 2/3 AD, BP= 2/3 FB, and CP= 2/3 CE.

13. The 30-60-90 Triangle Theorem VK R T X N M 30  SY Using the 30-60-90 Triangle Theorem, we get RS= 1/2 KR, TV= 1/2 KT, XY= 1/2 KX, and MN= 1/2 KN. After plugging in values given, you get RS= 1/2 (6), TV= 1/2 (10), XY= 1/2 (13), and MN= 1/2 (16). Therefore, RS=3, TV=5, XY=6.5, and MN=8. In ∆KMN,  M is a right angle and m  k=30. RS, TV, and XY are each perpendicular to KM. If KR=6, KT=10, KX=13, and KN=16, what are RS, TV, XY, and MN? What theorem did you use?

14. The Median Concurrence Theorem C EBA D P Using the Median Concurrence Theorem, we can find that AP= 2/3 AD and CP= 2/3 CE. By plugging in the numbers, you get AP= 2/3 (15) and CP= 2/3 (12). Therefore, AP=10 and CP=8. In the figure, AD and CE are medians. If AD=15 and CE=12, what are AP and CP?

15. quadrilateral-the union of 4 segments that intersect only at their endpoints sides-the 4 segments that make up the quadrilateral vertices-the endpoints of the segments that make up the quadrilateral convex-a quadrilateral is convex if no 2 of its vertices lie on opposite sides of a line containing a side of the quadrilateral

16. opposite sides-sides of a quadrilateral that never intersect opposite angles-angles that do not have a side of the quadrilateral in common consecutive sides-sides that have a common endpoint consecutive angles-angles that have a side of the quadrilateral in common diagonal-a segment in a quadrilateral that join 2 nonconsecutive vertices parallelogram-a quadrilateral in which both pairs of opposite sides are parallel

17. trapezoid-a quadrilateral in which only one pair of opposite sides are parallel bases-the parallel sides of the trapezoid median-the segment joining the midpoints of the nonparallel sides rhombus-a parallelogram all of which whose sides are congruent rectangle-a parallelogram all of whose angles are right angles square-a rectangle all of whose sides are congruent

18. Each diagonal separates a parallelogram into two congruent triangles. A BC D In the figure, given  ABCD is a parallelogram, then ∆BAD  ∆DCB. In a parallelogram, any two opposite sides are congruent. In the figure, given  ABCD is a parallelogram, AD=BC and AB=CD. A BC D

19. In a parallelogram, any two opposite angles are congruent. A BC D In the figure, given  ABCD is a parallelogram,  A  C and  B  D. Given a quadrilateral in which both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. A BC D In the figure, given AD=BC and AB=CD, then  ABCD is a parallelogram.

20. In a parallelogram, any two opposite angles are congruent. Given a parallelogram  ABCD with m  b=4x+15 and m  D=6x-27. Find the measures of the four angles. What theorem did you use? A D B C 6x-27 4x+15 Because we know that any two opposite angles are congruent in a parallelogram, we know that m  A=m  C and m  B=m  D. 6x-27=4x+15, 2x=42, and x=21, so m  B=m  D=99. Because  A is supplementary to  D and  B is supplementary to  C, m  A=m  C=81.

21. Given:  PQRS is a parallelogram and PW=PS and RU=RQ Prove:  SWQU is a parallelogram. R QP S W U SR 1.  PQRS is a parallelogram, PW=PS, RU=RQ 2. PS=RQ, SR=PQ 3. PW=RU 4. ∆SPR  ∆QRP 5.  SPW   QRU,  QPW   SRU 6. ∆SPW  ∆QRU 7. SW=QU 8. ∆PQW  ∆RSU 10.  SWQU is a parallelogram 9. QW=SU 1. Given 2. Opposite sides of a parallelogram are congruent 3. TPE 4. Diagonal separates parallelogram into 2  ∆’s 5. CPCTC 6. SAS 7. CPCTC 8. SAS 9. CPCTC 10. If both pairs of opposite sides are , then the quadrilateral Is a parallelogram Opposite sides are congruent, diagonal divides parallelogram into congruent triangles, if opposite sides are congruent, the quadrilateral is a parallelogram

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