LaSalle Preparatory School

Module 21 Hints & Help - Triangles

The triangle sum theorem states that the sum of the three angles inside a triangle is always 180.  If you know two angles, you can subtract them from 180 to find the third angle.  If the angles are given algebraically, set up an equation that follows the pattern ___ + ___ + ___ = 180 and solve for x.  Once you find the value of x, you can substitute the value in each expression and simplify to find the measure of each angle.

Vocabulary:

Interior Angle- any of the three angles inside the triangle

Exterior Angle- an angle on the outside of the triangle made by extending one of the sides of the triangle

Remote Interior Angles- the two angles that are inside the triangle and away from the exterior angle (they are not next to the exterior angle)

In this picture, the angle labled x° is an exterior angle, and the angles labled 37° and 72° are remote interior angles.

The exterior angle theorem states that an exterior angle is equal to the sum of the remote interor angles.  In the example above, the missing angle would be equal to 37° + 72°, or 109°.  If you know the measure of the exterior angle and one remote interior angle, you can subtract the two angles to find the other.  If the angles are given algebraically, set up an equation with the pattern interior angle 1 + interior angle 2 = exterior angle and solve for x.  Once you have the value of x, you can substitute it in each expression to find the measure of the angles.

Module 21 Hints & Help ~ Similarity

Similar- figures that have the same shape but not necessarily the same size.

The angle-angle (AA) similarity postulate states that if two angles of one triangle are congruent (equal) to two angles of another triangle, then the two triangles are similar.

Similar triangles help us with indirect measurement.  This is useful when we want to measure the height of something that can be hard to measure, like the height of a tree.  Because corresponding angles (same position but on different triangles) are congruent and corresponding sides are proportional in similar triangles, you can use similar triangles to solve real-world problems.

When writing your proportion be sure to make your comparissons consistent in both ratios!  For example, you may want to compare height and shadow like this:

       height of person    =   height of tree

       person's shadow         tree's shadow

Module 21 Hints and Help- Angles

When two parallel lines are cut by a third line (transversal) 8 angles are formed with two different measures.  The angles will either be congruent (equal) to each other or they will be supplementary (have a sum of 180).  If we are given the measure of one angle, we can quickly find the measure of the other seven by first finding the angles that are congruent to the given angle and then subtracting the given measure from 180 to find the others.  If the angles are measured algebraically, we can write and solve an equation by using the angle relationships.  If the angles are congruent, set the expressions equal to each other and solve for x; if the angles are supplementary follow the pattern ___ + ___ = 180 and solve for x.  

Remember that angles are named by 3 letters.  The middle letter is always the vertex (corner) point!

Vocabulary:

transversal- a line that cuts through two parallel lines

congruent- equal in measure

supplementary angles- two angle meaures that add up to 180; two angles that form a straight line

*vertical angles- angles that share a vertex but point in opposite directions (make an X shape)

*corresponding angles- angles in the same position, but on different parallel lines

*alternate interior angles- angles inside the parallel lines and on opposite sides of the transversal

*alternate exterior angles- angles that are outside the parallel lines and on opposite sides of the transversal

same side interior angles- angles that are inside the parallel lines and on the same side of the transversal

*these angle pairs are congruent

Here is a visual of the vocabulary that might help: parallel lines poster