Geometry Proofs

1. Metric Relationships
Showings Proofs and
Logical Thinking in Math

2. Proofs
 This topic deals with showing
logical steps as you try to
determine a length or angle in a
triangle or circle.
 Many of the concepts you
already know.

3. Congruency and Similarity
 Only if we can prove that two
triangles are congruent or
similar by (ASA, SAS, etc), we
can then determine the
lengths and angles of the
corresponding sides.

4. Complementary Angles
 Two angles are called complementary
angles if the sum of their degree
measurements equals 90 degrees. One
of the complementary angles is said to be
the complement of the other.
 N.B. These two angles can be "pasted"
together to form a right angle!

5. Supplementary Angles
 Two angles are called supplementary
angles if the sum of their degree
measurements equals 180 degrees. One
of the supplementary angles is said to be
the supplement of the other.
 Note that these two angles can be
"pasted" together to form a straight line!

6. Opposite Angles
 For any two lines that meet, such as in
the diagram below, angle AEB and angle
DEC are called opposite angles.
 Vertical angles have the same degree
measurement.

7. Interior Angles
 For any pair of parallel lines 1 and 2, that are
both intersected by a transversal, such as line 3
in the diagram below, angle A and angle C are
called interior angles.
 Interior angles are supplementary.
 < B and < D are also
 interior angles.

8. Corresponding Angles
 For any pair of parallel lines 1 and 2, that are
both intersected by a transversal, such as line 3
in the diagram below, < A and < C are called
corresponding angles.
 Corresponding angles are equal.

9. Alternate Interior Angles
 For any pair of parallel lines 1 and 2, that are
both intersected by a transversal, such as line 3
in the diagram below, < A and < D are called
alternate interior angles.
 Alternate interior angles have the same degree
measurement.

10. Alternate Exterior Angles
 For any pair of parallel lines 1 and 2, that are
both intersected by a transversal, such as line 3
in the diagram below, < A and < D are called
alternate exterior angles.
 Alternate exterior angles have the same degree
measurement.

11. Angle Bisector
 An angle bisector is a ray that divides an angle
into two equal angles.
 The blue ray on the right is the angle bisector of
the angle on the left.

12. Perpendicular Lines
 Two lines that meet at a right
angle are perpendicular.
 Like the corner of this
blackboard.

13. Pythagorean Theory
 In a right triangle, the square of the
hypotenuse is equal to the sum of
the squares of the other 2 sides.
 c2 = a2 + b2

14. Corollary to Pythagorean
Theorem
 IF the square of the length of
the longest side of a triangle
IS equal to the sum of the
squares of the lengths of the
other 2 sides, THEN it is a
right triangle.

15. Right Triangle Area
 The area of a triangle is
 A = bh
2
In a right triangle you can
usually find this 2 ways.

16. Products of the Sides Theorem
 In a right triangle, the product of the
length of the sides is equal to the
product of the lengths of the
hypotenuse and its altitude.
 ABxAC = BCxAD
2 2
 ABxAC =BCxAD
 or bh = BH

17. Geometric Mean
 The geometric mean of 2
numbers is the square root of
their product.
 E.g. 4, 9….4x9 = 36
 So the geometric mean is √36
 Which is 6.

18. Altitude to the Hypotenuse
Theorem
 The length of the altitude to
 the hypotenuse of a right
 triangle is the geometric
 mean between the lengths
 of the segments into which
 the altitude divides the
 hypotenuse.
 m2 = 3 x 7

19. Projection of a side
 A projection of a
 side of a triangle
 onto another side
 is similar to pushing
 one side vertically
 down onto another
 line.

20. Proportional Mean Theorem
 In a right triangle, each side is
the geometric mean between
the hypotenuse and that
side’s projection on the
hypotenuse.

21. 30˚ Theorem
 If a right triangle contains a 30˚ angle, the
side opposite the 30˚ angle is ½ the
length of the hypotenuse.

22. Median Theorem
 In a right triangle, the length of the
median to the hypotenuse is equal to ½
the length of the hypotenuse.

23. Angle Bisector Theorem
 In any triangle, the bisector of an angle
divides the opposite side into 2 segments
whose lengths are proportional those of
the adjacent sides.
 BD:DC = AB:AC

24. Exam Question

ABC is a right triangle in which segment AD measures 10 cm and segment DC, 25 cm.
A
B
C
D10 cm 25 cm
What is the measure of segment AB, to the nearest tenth?
A) 15.8 cm C) 22.5 cm
B) 18.7 cm D) 29.6 cm

25. Exam Question
Given the right triangle to the right.
AB
C
a
b
h
nm
c
Which of the following relations is true?
A) a  b = m  n C) h = m  n
B) a + b = c D) a2
= m  c

26. Exam Question
ABC is a right triangle in which segment AD measures 5 cm and segment DC, 10 cm.
A
B
C
D5 cm 10 cm
What is the measure of segment BD, to the nearest tenth?
A) 7.1 cm C) 8.1 cm
B) 7.6 cm D) 8.6 cm

27. Activity
 Page 34, Q. 1,2,7
 Page 46, Q. 1, 2, 5, 10,12
 Page 58, Q. 1, 2, 4, 12
 Page 68, Q. 2, 4,6