ksu #gsp for hw 4 432043 yes 397469 374787 445707 yes 441983 483280 yes 394078 yes 312014 yes 406936...
TRANSCRIPT
KSU # GSP for HW 4
432043 YES
397469374787445707 YES
441983483280 YES
394078 YES
312014 YES
406936 YES
443837339238401035420456
KSU # GSP for HW 4
350057 YES
428483338465340871 YES
427222 YES
427055341525493249 YES
401794449724437486 YES
382516 YES
414398 YES
If you do not have YES next to your student #, either you did not send me GSP constructions or I am waiting for corrections.
1. Construct an angle which is the complement of angle A.
B
C
A
1. Construct an angle which is the complement of angle A.
A
B
C
P
B
C
A
P R
P
S
Q
RP
From HW # 4
A
B
C
D
A
B C
P
Q
From HW # 4
A
B C
P
Q
From HW # 4
A
B C
P
Q
From HW # 4
Q
P
A B
Q
P
A B3. APQ BPQ
Outline of proof of the construction
1. AP AQ BP BQ
2. PQ PQ
4. APQ BPQ
A
P
Q
P
B
Q
M
Q
P
A BM
A
P
Q
P
B
Q
3. APQ BPQ
4. APQ BPQ
Outline of proof of the construction
1. AP AQ BP BQ
2. PQ PQ
6. APM BPM
8. AMP is supplementary to BMP
9. AMP and BMP are right angles
10. PQ is the perpendicular bisector of AB
5. PM PM
7. AM BM , AMP BMP
7. True or False: The three angles shown at the right are supplementary. 8. True or False: Triangle ABC shown is a right triangle. 9. In the diagram, CBD is a right angle, and
CA and DA bisect angles BCD and BDC, respectively. What is the measure of CAD? Explain why your answer must be correct. 10. In the diagram, DABD , and AD bisects BAC. What is the measure of DBA?
A
B
C
D
12
38
C
B D
A
47 64
69
A B
C
x 4x
5x
False
True
C
B D
A
9. In the diagram, CBD is a right angle, and bisect angles BCD and BDC, respectively. What is the measure of CAD?
CA
135°
DA
B
A
C
D
12
38
10. In the diagram, , and bisects ABC.
What is the measure of DBA?
DABD AD
50°
P
Q
A
B
C
The sum of the measures of the angles of a triangle is 180°.
The existence of this parallel is based on Euclid’s fifth postulate, (the parallel postulate).
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn …to the line.
• If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
In the diagram, ABD DCA and . If the
measure of DCB is 50, what is the measure of A?
BCBD
A
B C
D
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn …to the line.
• If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn …to the line.
• If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
• If one side of a triangle is extended, then the measure of the exterior …angle(s) formed is equal to the sum of the measures of the two remote …interior (non-adjacent interior) angles.
If one side of a triangle is extended, then the measure of the exterior angle(s) formed is equal to the sum of the measures of the two remote interior (non-adjacent interior) angles.
One exterior angle at each vertex of ABC is drawn.
What is the sum of the measures of these three angles?
A B
C
The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2)180o.
The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2)180o.
The sum of the measures of the interior angles of a convex polygon with n sides is (n – 2)180o.
In this nine sided figure, all 6 diagonals from the same vertex form 7 triangles (9-2).
A
The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360o.
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn …to the line.
• If two angles of one triangle are congruent to two angles of another …triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
• If one side of a triangle is extended, then the measure of the exterior …angle(s) formed is equal to the sum of the measures of the two remote …interior (non-adjacent interior) angles.
• The sum of the measures of the interior angles of a convex polygon with …n sides is (n – 2)180o.
• The sum of the measures of the exterior angles of any convex polygon, …one angle at each vertex, is 360o.
Practice Problems
1. Find the sum of the measures of A, B, C, D, and E
A
B
C
DE
40
2. In the diagram, bisects ABC. What is the measure of BED?
A
D
B
C
E
108
96
48
BE
Practice Problems
1. Find the sum of the measures of A, B, C, D, and E
A
B
C
DE
40
2. In the diagram, bisects ABC. What is the measure of BED?
A
D
B
C
E
108
96
48
BE
102
460
3. In the diagram, compute the sum of the angles numbered 1 through 8. 1
2
34
5 6
7
8
4. In quadrilateral ABCD, the measures of angles A, B, C and D, in that order, form an increasing arithmetic sequence. Explain why sides AB and CD must be parallel.
3. In the diagram, compute the sum of the angles numbered 1 through 8. 1
2
34
5 6
7
8
4. In quadrilateral ABCD, the measures of angles A, B, C and D, in that order, form an increasing arithmetic sequence. Explain why sides AB and CD must be parallel.
720
A convex polygon with n-sides has .2
3)(n)(n
A convex polygon with n-sides has .2
3)(n)(n
Homework:
Download, print, and complete Homework # 5