Prove Area Theorem Of Similar Triangles
Answers
Angle - Angle (AA)
Side - Angle - Side (SAS)
Side - Side - Side (SSS)
Corresponding Angles
In geometry, correspondence means that a particular part on one polygon relates exactly to a similarly positioned part on another. Even if two triangles are oriented differently from each other, if you can rotate them to orient in the same way and see that their angles are alike, you can say those angles correspond.
The three theorems for similarity in triangles depend upon corresponding parts. You look at one angle of one triangle and compare it to the same-position angle of the other triangle.
Proving Triangles Similar
Here are two congruent triangles. To make your life easy, we made them both equilateral triangles.
identical equilateral triangles
△FOX is compared to △HEN
. Notice that ∠O on △FOX corresponds to ∠E on △HEN
. Both ∠O and ∠E are included angles between sides FO and OX on △FOX
, and sides HE and EN on △HEN
.Side FO is congruent to side HE; side X is congruent to side EN, and ∠O and ∠E are the included, congruent angles.
The two equilateral triangles are the same except for their letters. They are the same size, so they are identical triangles. If they both were equilateral triangles but side EN was twice as long as side HE, they would be similar triangles.
Angle-Angle (AA) Theorem
Angle-Angle (AA) says that two triangles are similar if they have two pairs of corresponding angles that are congruent. The two triangles could go on to be more than similar; they could be identical. For AA, all you have to do is compare two pairs of corresponding angles.
Trying Angle-Angle
Here are two scalene triangles △JAM and △OUT
. We have already marked two of each triangle's interior angles with the geometer's shorthand for congruence: the little slash marks. A single slash for interior ∠A and the same single slash for interior ∠U mean they are congruent. Notice ∠M is congruent to ∠T because they each have two little slash marks.
Since
∠A is congruent to ∠U, and ∠M is congruent to ∠T, we now have two pairs of congruent angles, so the AA Theorem says the two triangles are similar.
similar triangles AA theorem
Tricks of the Trade
Watch for trickery from textbooks, online challenges, and mathematics teachers. Sometimes the triangles are not oriented in the same way when you look at them. You may have to rotate one triangle to see if you can find two pairs of corresponding angles.
Another challenge: two angles are measured and identified on one triangle, but two different angles are measured and identified on the other one.
Because each triangle has only three interior angles, one each of the identified angles has to be congruent. By subtracting each triangle's measured, identified angles from 180°, you can learn the measure of the missing angle. Then you can compare any two corresponding angles for congruence.
Side-Angle-Side (SAS) Theorem
The second theorem requires an exact order: a side, then the included angle, then the next side. The Side-Angle-Side (SAS) Theorem states if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar.
Trying Side-Angle-Side
Here are two triangles, side by side and oriented in the same way. △RAP and △EMO both have identified sides measuring 37 inches on △RAP and 111 inches on △EMO, and also sides 17 on △RAP and 51 inches on △EMO
. Notice that the angle between the identified, measured sides is the same on both triangles:
47°
.similar triangles SAS theorem
Is the ratio 37/111 the same as the ratio 17/51? Yes; the two ratios are proportional, since they each simplify to 1/3
. With their included angle the same, these two triangles are similar.
Side-Side-Side (SSS) Theorem
The last theorem is Side-Side-Side, or SSS. This theorem states that if two triangles have proportional sides, they are similar. This might seem like a big leap that ignores their angles, but think about it: the only way to construct a triangle with sides proportional to another triangle's sides is to copy the angles.
Trying Side-Side-Side
Here are two triangles, △FLOand △HIT
. Notice we have not identified the interior angles. The sides of △FLO
measure 15, 20 and 25 cms in length. The sides of △HIT measure 30, 40 and 50 cms in length.similar triangles SSS theoremYou need to set up ratios of corresponding sides and evaluate them:
They all are the same ratio when simplified.
. So even without knowing the interior angles, we know these two triangles are similar, because their sides are proportional to each other.
Note:
You also can apply the three triangle similarity theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS) or Side - Side - Side (SSS), to determine if two triangles are similar.