The document contains proofs of several geometric theorems: 1) Intersecting lines L and K have vertically opposite angles 1 and 2 that are equal. 2) The interior angles of any triangle sum to 180 degrees. 3) The exterior angle of a triangle is equal to the sum of the two interior opposite angles. 4) In an isosceles triangle, the base angles are equal. 5) Corresponding angles are equal when two parallel lines are intersected by a transversal. 6) Angles opposite equal sides of a triangle are equal.

1. L
K
3
1 2

2. Given:
To prove :
Intersecting lines L and K, with
vertically opposite angles 1 and 2.
Construction:
1=2
Label angle 3
Proof: Straight angle
Straight angle
1+3=180
2+3=180
 1+3=3+2
 1=2
.....Subtract 3 from both
sides

3. 4
5
a
3
1 2
b c
.
5

4. Given:
The triangle abc with 1,2 and 3.
To Prove: 1+2+3=180
Construction:
Proof:
Draw a line through a, Parallel to bc.
Label angles 4 and 5.
1=4 and
2=5
Alternate
angles
1+2+3=4+5+3
But
4+5+3=180
Straight
angle
 1+2+3=180
Angle sum property

5. a
3
1
2 4
b c
Q.E.D.

6. Given: A triangle with interior opposite angles 1 and 2 and the
exterior angle 3.
To prove: 1+ 2= 3
Proof: 1+ 2+ 4=180
3+ 4=180
Angle sum property
 1+ 2+ 4= 3+ 4
Straight angle
 1+ 2= 3

7. a
3 4
b 1 2 c
d

8. Given: The triangle abc, with ab = ac and base angles 1 and 2.
To prove: 1 = 2
Construction: Draw ad, the bisector of bac. Label angles 3
and 4.
Proof:
consider ABD and ACD
ab = ac
construction
3 = 4 given
ad = ad common
SAS criteria
 ABD ACD
 1 = 2 CPCT

9. 1
3
2
t
A B
P
C D
Q

10. Given: AB||CD and EFGH is a transversal.
To Prove:  1 =  2
Proof : 1= 5 [1] v .o .a
5= 2 [2] corresponding
angles
from equation 1 and 2 we get ,
1= 2

11. Angles opposite to equal
sides of triangle are
equal A
B
D C

12. Given: AB=AC
To prove: B= C
Construction : Draw the bisector AD of A which meets BC at D
Proof: In ABD and ACD
AB = AC g i v e n
BAD= CAD By construction
AD=AD Common
=> ABD =
ACD
SAS criteria
=> B = C By C.P.C.T.