Proofs In Geometry

Proofs In Geometry

The Stuff Geometry is Known For!

Geometry Terms

• Postulate - a statement we accept without proof

• Theorem - a mathematical statement that has been proven true or needs to be proven

• Proof - the process of proving or disproving a statement

• A theorem often takes the form of an if - then statement, if this is true, then this has to be true

Let's start with these postulates

• The interior angles in a triangle sum to 180°

• The sum of an interior and exterior angle of a polygon is equal to 180°

• The sum of the the exterior angles of polygon is equal to 360*

• Only one line can be drawn between two points

• Parallel lines never intersect

Multiple Choice

The exterior angle of this regular polygon is 45°

• False: The sum of the exterior angles of a polygon is 180›, there are 8 exterior angles and 8 x 45 = 360°

• True, the sum of the exterior angles in a polygon is 360°, there are 8 exterior angles and 8 x 45 = 360°

• We cannot assume this polygon is regular, so it may have more or less than 360°

• False, you can tell by inspection that the exterior angle is obtuse, so it is over 90›

Multiple Choice

The exterior angle in the 5 sided regular polygon is 72°, the interior angle must equal 118°

• True, it is a postulate that the interior angle and exterior angle equal 180›

• False, the sum of the interior and exterior angle of a polygon equals 180°

• False, the exterior angles must equal 360°

• False, all exterior angles are straight lines so they equal 180°

Multiple Choice

These two lines will intersect. Select the best answer

• True, it is not shown in the picture

• Maybe, you need to see more of the picture

• False, parallel lines never meet or intersect

• False, it was given that these are parallel lines, thus they will never meet by the parallel line postulate

Multiple Choice

Angle "a" equals 60°. Is this statement true and why?

• False: it is equal to 70°, it is an isosceles triangle

• False: it is equal to 50°, it is an isosceles triangle

• True, the sum of the angles must equal 180°

• False, the sum of the angles must equal 360°

Multiple Choice

You can draw more than one line that connect these two points

• True, you can draw many lines of different colors

• False, a basic postulate is that there is only one line possible between two points

• True, you can draw parallel lines between these two points

• True, you can draw intersecting lines between these two points

How did you do?

These were simple proofs, you only needed to apply one of the postulates to prove a statement true or false. In Geometry we often have to prove a statement true or false using many postulates or theorems that have already been proven.

Multiple Select

Which of these are true of a postulate - there is more than one.

• We must prove them to be true

• We accept them to be true

• They are true because they are obvious, like the meaning of the term "point".

• Two lines intersect at only one point would be an example of a postulate

Fill in the Blanks

The image shown is an example of a postulate or theorem

Why was the previous question a postulate and not a theorem

We accept as obvious that those two straight lines will never cross each other at more than one point - you can see that in the drawing. So we accept it - thus it is a postulate

Isosceles Triangle

An isosceles triangle has two equal sides. We do not need to prove that, that is a definition. So it is like a postulate.

The statement that angles A and C are equal however must be proven. We cannot assume that. How can we prove that?

Watch this video on how to prove that the angles opposite the equal sides are equal.

Multiple Choice

In this proof, what definition did we accept as true - note that both of these definitions are true, which one did the video use

• that the midpoint of a line cuts the line into equal line segments

• that the angle bisector cuts and angle into two equal angles

Isosceles Triangle theorem

• We add this to the list of proven theorems

• You can use this theorem to solve problems and no longer would have to prove this correct

• The equal angles in an isosceles triangle are called the "base" angles.

Multiple Choice

Using the isosceles triangle theorem if angle D=50°, what is the measure of angle F?

• 50°, because of the isosceles triangle theorem

• 40°, the base angles of an isosceles triangle are complementary - they add up to 90°

• 130°, the angles in a triangle add up to 180°

Multiple Choice

Given the triangle shown is an isosceles triangle, angle B has a measure of 30°, what is the measure of angle A and C?

• 150°, the angles add up to 180°

• 75° we can prove as follows

  The sum of all the angles in a triangle = 180°

  Thus the sum of angle A + C = 150° (180-30)

  Angle A = Angle C by the isosceles triangle theorem

  Thus Angle A and Angle C = 75° (150/2)

• I could measure the angles, but since I can calculate them as shown I will calculate them - please don't pick me for the answer

Multiple Choice

What is the value of the exterior angle of this equilateral triangle?

• 60°, because the angles are equal and the three angles must equal 180°

• 180° - the exterior angle is supplementary to the interior angle

• 120°, the exterior angle is supplementary to the interior angle which is 60°

Multiple Choice

Which of these statements is a theorem

• The sum of the interior angles in a triangle sum to 180°

• Any three points define a plane

• The angles opposite the congruent sides of an isosceles triangle are equal

• Parallel lines never intersect