Guided Practice

Press on the numbers to see the steps.

1

Lines AB and CD intersect at point E.

And m∠CEA + m∠AED = 180°.

And m∠CEA + m∠AED = 180°.

2

Similarly, m∠AED + m∠DEB = 180°,

because those angles also form a linear pair.

because those angles also form a linear pair.

3

This means that:

m∠CEA + m∠AED = m∠DEB + m∠AED.

m∠CEA + m∠AED = m∠DEB + m∠AED.

4

If a + b = c + b, then a = c.

So, m∠CEA = m∠DEB.

So, m∠CEA = m∠DEB.

5

∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB

Start Over

Identify all the pairs of vertical angles below and then write the angle measure.

and are vertical angles.

and are vertical angles.

The measure of Angle 4 is °.

Identify all the pairs of vertical angles below and then write the angle measure.

and are vertical angles.

and are vertical angles.

The measure of Angle 6 is °.

Identify all the pairs of vertical angles below and then write the angle measure.

and are vertical angles.

and are vertical angles.

The measure of Angle b is °.

Lines m and n are parallel. Identify one pair of corresponding angles.

Angles and are corresponding angles.

Lines m and n are parallel. Identify one pair of alternate interior angles.

Angles and are alternate interior angles.

Lines m and n are parallel. Identify one pair of alternate exterior angles.

Angles and are alternate exterior angles.

Lines m and n are parallel. Identify one pair of alternate exterior angles.

Angles and are alternate exterior angles.

Lines m and n are parallel. Identify one pair of corresponding angles.

Angles and are corresponding angles.

Lines m and n are parallel. Identify one pair of alternate interior angles.

Angles and are alternate interior angles.

Press on the numbers to see the steps.

1

Start with △ABC. Extend BC to point D.

Line BD will be a transversal.

Line BD will be a transversal.

2

Draw line EC parallel to segment AB.

Draw transversal AC.

Draw transversal AC.

3

Corresponding angles are congruent.

m∠ECD = m∠ABC

m∠ECD = m∠ABC

4

Alternate interior angles are congruent.

m∠ECA = m∠CAB

m∠ECA = m∠CAB

5

m∠A + m∠B + m∠C = 180°.

A triangle's angle measures sum to 180°.

A triangle's angle measures sum to 180°.

Start Over

Determine the unknown angle measure.

The measure of Angle X is °.

Determine the unknown angle measure.

The measure of Angle Q is °.

Determine the unknown angle measure.

The measure of Angle C is °.

Angles A and C are congruent. Determine the measure of each angle.

Angles A and C each measure °.

Triangles are called 'similar' if their corresponding side lengths form equal ratios with each other and/or their corresponding angles are congruent. In the image below, BC is parallel to DF.

If AB/AD = BC/DF = AC/AF . . .

and/or m∠B = m∠D, m∠C = m∠F, m∠A = m∠A . . .

Then, △ABC ~ △ADF. The ~ symbol is read as 'is similar to.'

Since the sum of the angle measures of a triangle is always equal to 180°, we need to know only that two pairs of corresponding angles are congruent to determine if two triangles are similar. This is called Angle-Angle Similarity, or AA Similarity.

△PQR ~ △STV

When two triangles are similar, then all of their corresponding angle pairs are congruent and all of their corresponding side length ratios are equal. Similar triangles are named with their corresponding parts in the same order.

For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.

At each vertex, there are two possible exterior angles, E, which are congruent because they are vertical angles.

In each case, m∠E = m∠R_{1} + m∠R_{2}.

Line segments BC and DF are parallel. Determine the relationship between angles ABC and ADF and between angles BCA and DFA. Write corresponding, alternate interior, or alternate exterior.

∠ABC and ∠ADF are angles.

∠BCA and ∠DFA are angles.

And ∠A ≅ ∠A, so all three pairs of corresponding angles are congruent and △ABC and △ADF are similar.

Lines m and n are parallel, and lines s and t are parallel. Show that △JDF ~ △DVR.

∠FJD ≅ ∠

∠FDJ ≅ ∠

So, △JDF ~ △DVR.

Line segment TV is parallel to line segment QR (TV || QR). Show that △XVT is similar to △XRQ. Then determine the length of TV.

∠QRX ≅ ∠

∠XQR ≅ ∠

The length of TV is units.

Determine the lengths of BC and DC. Write the answers as fractions.

The length of BC is units.

The length of DC is units.

Lines a and b are parallel. Identify the congruent angles and complete the congruence statement.

Alternate interior angles: ∠NFC and ∠

△ is similar to △.

Determine the measure of Angle T.

The measure of Angle T is °.

Determine the measures of the angles.

The measure of ∠BCA is °.

The measure of ∠CAB is °.

The measure of ∠CBD is °.

Determine the measures of the angles.

The measure of ∠DCE is °.

The measure of ∠DEM is °.

What is the sum of any three exterior angle measures of a triangle?

The sum of 3 different exterior angle measures of a triangle is °.

You can use what you know about Angle-Angle Similarity, the ratios of side lengths of similar figures, corresponding angles, alternate interior angles, alternate exterior angles, vertical angles, exterior angles, and the sum of the measures of the interior angles of a triangle to solve all kinds of problems.

Try the problems in this section. Don't give up!

Quadrilateral ADEF is a square, so DE and AC are parallel, and EF and BA are parallel. Determine the length of AD as a decimal.

AD has a length of units.

The rays shown are parallel. Determine the measure of ∠m.

The measure of ∠m is °.

Line m is parallel to line n, and line p is parallel to line q. If the measure of ∠1 is x°, how many other labeled angles also measure x°?

other angles measure x°.

Parallel line segments are marked. Determine the measures of angles v and w.

The measure of ∠v is °.

The measure of ∠w is °.

In an isosceles triangle, the angles opposite the congruent sides are congruent angles. In this diagram, PQ = PS = RS. Determine the measure of ∠QPR.

The measure of ∠QPR is °.

Click and drag to draw polygons using the dots as vertices. When you form a closed polygon, the dots that you cross, which are called boundary points, are circled.

Count the number of circled boundary points and divide by 2. Then, add this number to the number of dots on the inside of the polygon. And then subtract 1. The result is the area of the figure!

Try all kinds of different polygons. Does the area formula always work?

Consider the diagram. Angles 1 and 2 are vertical angles, so they are congruent. And angles 3 and 4 are vertical angles, so they are congruent.

Angles 1 and 3 are corresponding angles, angles 2 and 3 are alternate interior angles, and angles 1 and 4 are alternate interior angles.

Name another pair of labeled corresponding angles, besides angles 1 and 3.

Angles and are also corresponding angles.