Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK
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3. −−<br />
BX −−<br />
XC 3. Definition of a<br />
median.<br />
4. −−−<br />
AX −−−<br />
AX 4. Reflexive<br />
property of<br />
congruence.<br />
5. ABX ACX 5. SSS SSS.<br />
6. ABX ACX 6. Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.<br />
7. Construct a second<br />
median, −−<br />
BY ,<br />
from vertex B.<br />
7. A median of a<br />
triangle is a line<br />
segment with<br />
one endpoint<br />
at any vertex<br />
of <strong>the</strong> triangle,<br />
extending to <strong>the</strong><br />
midpoint of <strong>the</strong><br />
opposite side.<br />
8. −−<br />
BA −−<br />
BC 8. Definition of an<br />
equilateral<br />
triangle.<br />
9. −−<br />
AY −−<br />
YC 9. Definition of a<br />
median.<br />
10. −−<br />
BY −−<br />
BY 10. Reflexive<br />
property of<br />
congruence.<br />
11. BAY BCY 11. SSS SSS.<br />
12. BAY BCY 12. Corresponding<br />
angles of congruent<br />
triangles are<br />
congruent.<br />
13. ABC is equiangular.<br />
13. BCY is <strong>the</strong><br />
same as ACX,<br />
and all three<br />
angles of <strong>the</strong> triangle<br />
are equal.<br />
5 mA mC 3x 5<br />
mA mB mC 180<br />
(3x 5) (4x 10) (3x 5) 180<br />
10 x 180<br />
x 18<br />
mC 3(18) 5 59<br />
6 The measure of each angle is 60. So,<br />
3x 6 60; x 22. And 3y 6 60;<br />
y 18.<br />
7 Show that RXZ TYZ SYX by<br />
SAS SAS, and −−<br />
XY −−<br />
YZ −−<br />
ZX because<br />
corresponding parts of congruent triangles<br />
are congruent.<br />
8 Construct −−<br />
AC and −−<br />
BD , and label <strong>the</strong>ir intersection<br />
E. ABE CBE. −−<br />
AC and −−<br />
BD are<br />
perpendicular. AED and CDE are right<br />
angles; <strong>the</strong>re<strong>for</strong>e, AED CDE.<br />
−−<br />
AE −−<br />
CE . −−<br />
ED −−<br />
ED . AED CED.<br />
CAD ACD.<br />
9 mA mB mC 180<br />
(6x 12) (8x 8) (3x 6) 180<br />
17x 1 0 180<br />
x 10<br />
mA 72; mB 72; mC 36<br />
The triangle is isosceles.<br />
10 mB 180 2x<br />
5-4 Working With Two<br />
Pairs of Congruent<br />
Triangles<br />
(pages 75–76)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. ABC EFG<br />
2. −−<br />
AC −−<br />
EG<br />
3. −−<br />
BD is a median.<br />
4. D is <strong>the</strong> midpoint (Definition of<br />
of −−<br />
AC . median)<br />
5. DC 1 _ AC (Definition of mid-<br />
2<br />
point)<br />
6. −−<br />
FH is a median.<br />
7. H is <strong>the</strong> midpoint of −−<br />
EG .<br />
8. HG 1 _ EG<br />
2<br />
9. AC EG<br />
10. HG 1 _ AC<br />
2<br />
11. HG DC<br />
5-4 Working With Two Pairs of Congruent Triangles 21
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