Triangle Similarity and Congruence

Two figures of the same shape carry the same angles and proportional sides, and that one fact converts a length you know into a length you want.

Two figures can be related in two strengths. Congruent figures are identical: same shape and same size, so every matching part is equal. Similar figures are the same shape at possibly different sizes; one is a scaled copy of the other. Similarity is the workhorse, because it lets a figure you can measure stand in for one you cannot: knowing the shape is preserved, a single proportion turns a known length into an unknown one. Almost every question in this section reduces to recognizing a pair of similar triangles and writing the proportion that links them.

▸ Distinguish congruent from similar triangles

In congruent figures, corresponding sides and corresponding angles are all equal; one can be slid, turned, or flipped onto the other exactly. In similar figures, corresponding angles are still equal, but corresponding sides are merely proportional: each side of one is the same constant multiple, the scale factor, of the matching side of the other. Congruence is the special case of similarity in which the scale factor is 1.

▸ Establish triangle similarity with angle-angle

For triangles, similarity is unusually easy to establish. Because the angles of a triangle must sum to 180 degrees, knowing two pairs of equal angles forces the third pair to match as well, so two triangles with two pairs of equal angles are automatically similar. This is the AA (angle-angle) criterion, and it is the tool you will reach for most, because equal angles are everywhere: shared angles, vertical angles, and the equal angles created by parallel lines all hand you the pairs you need.

FIGURE 1 Similar triangles have equal corresponding angles and proportional corresponding sides. Here every side of the larger is twice its match.

Once two triangles are known to be similar, their corresponding sides are locked into a single ratio. Writing that ratio as an equation, being careful to match each side with its true counterpart, is what produces the answer.

▸ Use similarity proportions to find missing lengths

The mechanics are always the same: identify which sides correspond, set their ratios equal, and solve. The one discipline that matters is correspondence, pairing each side with the side that plays the same role in the other triangle, which the equal angles tell you. Get the pairing right and the proportion solves itself.

WORKED Solve for a side by proportion

Triangle ABC is similar to triangle DEF, with AB corresponding to DE. Given AB = 8, BC = 10, and DE = 12, find EF.

Corresponding sides share one ratio, so AB/DE = BC/EF:

8/12 = 10/EF → 8 × EF = 120 → EF = 15.

The shape is fixed; only the size scales, here by a factor of 12/8 = 1.5 from ABC to DEF.

WORKED Measure indirectly with shadows

A 5-foot person casts a 3-foot shadow while a flagpole casts a 24-foot shadow at the same time. The sun's rays make equal angles, so the two right triangles are similar.

height / 24 = 5 / 3 → height = 24 × (5/3) = 40 feet.

Indirect measurement is just similar triangles applied to a real-world setup.

▸ Use the altitude to the hypotenuse to relate three similar triangles

Drawing the altitude from the right angle of a right triangle down to the hypotenuse splits the original triangle into two smaller right triangles. All three triangles, the original and the two pieces, share their angles pairwise, so all three are similar to each other by angle-angle. That shared similarity forces a proportion between the two segments the altitude creates and the altitude itself: if the altitude splits the hypotenuse into pieces of length p and q, the altitude h satisfies h² = pq, the altitude is the geometric mean of the two segments. This is not a separate fact to memorize; it is the same similar-triangle proportion used everywhere else in this chapter, applied to a specific figure.

WORKED Find an altitude from the segments it creates

In a right triangle, the altitude to the hypotenuse divides it into segments of length 4 and 9. What is the length of the altitude?

Apply the relationship directly: h² = 4 × 9 = 36, so h = 6.

▸ Use proportional segments created by a line parallel to a triangle side

One configuration appears so often it deserves its own recognition: a line drawn parallel to one side of a triangle, cutting the other two sides. Because the parallel line creates equal corresponding angles, it carves off a smaller triangle similar to the whole, and the two sides it crosses are divided in the same ratio.

FIGURE 2 A line parallel to one side cuts the other two sides proportionally, and creates a smaller triangle similar to the original.

WORKED Use a parallel side-divider

In triangle ABC, D lies on AB and E on AC with DE parallel to BC. If AD = 6, DB = 4, and AE = 9, find EC.

The parallel line splits the two sides in equal ratios, AD/DB = AE/EC:

6/4 = 9/EC → 6 × EC = 36 → EC = 6.

A special case is the midsegment: a segment joining the midpoints of two sides is parallel to the third and exactly half its length.

▸ Recognize similarity in a bow-tie configuration

A second configuration is easy to miss because the two triangles overlap rather than nest. Two segments cross at a point, forming an X, or bow tie. If the two outer segments are parallel, the crossing point creates vertical angles, and the parallel lines create equal alternate-interior angles, so the two triangles are similar by AA even though they point in opposite directions.

FIGURE 2B Segments AD and BC cross at E. AB is parallel to CD, so triangle ABE is similar to triangle DCE, even though the triangles point away from each other.

WORKED Solve a bow-tie similarity

Segments AD and BC intersect at E, with AB parallel to CD. If AE = 5, ED = 10, and AB = 8, what is CD?

The vertical angles at E and the parallel sides give triangle ABE similar to triangle DCE, with AE and ED the corresponding sides that set the ratio.

AE / ED = AB / CD → 5 / 10 = 8 / CD → CD = 16

The two triangles share only the point E; nothing about them nests or overlaps in area the way the parallel-side-divider configuration does, which is exactly why the bow tie is easy to miss. Checking that the two parallel sides, not the two crossing segments, are the ones set proportional is the detail this configuration is built to test.

▸ Apply linear and squared scale factors to perimeter and area

Scaling a figure changes its measurements in a patterned way that the test loves to probe. Lengths (sides, perimeters, diagonals) scale by the factor k itself. But area scales by k², because area is a product of two lengths, and volume, a product of three, scales by k³. So doubling every side doubles the perimeter but quadruples the area.

FIGURE 3 Lengths scale by k, but area scales by k squared: doubling the side multiplies the area by four.

HOW SIMILARITY SCALES MEASUREMENTS

Lengths (sides, perimeter): multiply by the scale factor k.

Areas: multiply by k².

Volumes: multiply by k³.

Reversing this: if areas are in ratio a : b, the sides are in ratio √a : √b.

▸ Determine when given information proves similarity or congruence

Some geometry questions do not ask for a length or an angle at all; they ask which additional fact would be enough to force a relationship, a pair of triangles similar, a quadrilateral a parallelogram, a triangle isosceles. Treat each answer choice as a proposed condition and ask what it unlocks, not whether it merely looks consistent with the figure.

Similarity needs only two pairs of equal angles, the AA criterion already in hand. Congruence is stricter: it forces both the shape and the size to match, so it takes a full side measurement, not just an angle. Five combinations are guaranteed sufficient: SSS (three pairs of equal sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a non-included side), and HL for right triangles specifically (matching hypotenuse and one leg). Two combinations that look similar are traps: AAA proves similarity only, since it fixes shape but not size, and SSA proves neither: two sides and a non-included angle can describe two genuinely different triangles, so it forces nothing on its own. Parallel lines crossed by a transversal create the equal angle pairs that often supply the angle piece of any of these conditions.

WORKED Test a proposed condition

In triangle ABC, D lies on AB and E lies on AC. Which additional fact is sufficient to conclude that DE is parallel to BC?

The condition that actually forces the parallel relationship is AD/DB = AE/EC, the side-splitting ratio from earlier in this chapter; matching that ratio is what guarantees DE and BC never meet at a different angle.

A choice claiming DE and BC merely “look parallel” in the figure, or that some unrelated pair of sides is equal, is not sufficient: it may be consistent with the drawing without forcing the relationship the way a genuine condition must.

The habit to build is refusing appearance as evidence. A pair of sides that looks equal, an angle that looks right, a point that looks centered: none of these count unless the problem states them or an earlier step derives them. A sufficiency question is really asking whether a fact proves a relationship or merely happens to be drawn alongside it.

▸ Practice: similarity and congruence

1. Two similar triangles have corresponding sides in the ratio 3 : 5. The smaller triangle has a perimeter of 24. What is the perimeter of the larger triangle? (grid-in)

2. Triangle ABC is similar to triangle DEF, with AB corresponding to DE. If AB = 8, BC = 10, and DE = 12, what is the length of EF? (grid-in)

3. In triangle ABC, point D lies on AB and point E lies on AC with DE parallel to BC. If AD = 6, DB = 4, and AE = 9, what is the length of EC? (grid-in)

4. A 5-foot-tall person casts a 3-foot shadow at the same time that a flagpole casts a 24-foot shadow. What is the height of the flagpole, in feet? (grid-in)

6. In a right triangle, the altitude drawn to the hypotenuse divides it into segments of lengths 4 and 9. What is the length of that altitude? (grid-in)

7. Two similar triangles have areas in the ratio 25 : 4. A side of the larger triangle has length 15. What is the length of the corresponding side of the smaller triangle? (grid-in)

8. In triangle ABC, D and E are the midpoints of sides AB and AC. If BC = 14, what is the length of segment DE? (grid-in)

9. Two similar triangles have corresponding sides of length 12 and 18. If the area of the smaller triangle is 40, what is the area of the larger triangle? (grid-in)

10. A scale model of a tower is built at a scale of 1 to 50. The actual tower is 75 meters tall. What is the height of the model, in centimeters? (grid-in)

11. In triangle ABC, AB = 15 and AC = 20. Point D on AB and point E on AC satisfy DE parallel to BC. If AD = 9, what is the length of AE? (grid-in)

12. Triangles ABC and ADE share vertex A, with D on AB and E on AC, and DE parallel to BC. If AD = 6, AB = 15, and DE = 8, what is the length of BC? (grid-in)

13. Triangle PQR is similar to triangle ABC with a scale factor of 2, where PQR is the larger. If the area of triangle ABC is 12, what is the area of triangle PQR? (grid-in)

▸ Solutions: similarity and congruence

1: 40. Perimeters scale like sides, in the ratio 3 : 5. So the larger perimeter is 24 × (5/3) = 40.

2: 15. Corresponding sides share a ratio: AB/DE = BC/EF, so 8/12 = 10/EF, giving EF = 120/8 = 15.

3: 6. A line parallel to BC splits the sides proportionally: AD/DB = AE/EC, so 6/4 = 9/EC and EC = 36/6 = 6.

4: 40. The sun creates similar right triangles, so height over shadow is constant: h/24 = 5/3, giving h = 40 feet.

6: 6. The altitude to the hypotenuse is the geometric mean of the two segments it creates: √(4 × 9) = √36 = 6. This follows from the two smaller triangles being similar.

7: 6. The side ratio is the square root of the area ratio: √25 : √4 = 5 : 2. So the smaller side is 15 × (2/5) = 6.

8: 7. The segment joining two midpoints is a midsegment, parallel to the third side and half its length: 14/2 = 7.

9: 90. Sides are in ratio 12 : 18 = 2 : 3, so areas are in ratio 4 : 9. The larger area is 40 × (9/4) = 90.

10: 150. The model is 1/50 of the tower: 75/50 = 1.5 meters, which is 150 centimeters.

11: 12. With DE parallel to BC, AD/AB = AE/AC, so 9/15 = AE/20, giving AE = 20 × (9/15) = 12.

12: 20. Since DE is parallel to BC, triangle ADE is similar to triangle ABC by angle-angle. The scale factor from the small triangle to the large one is AB/AD = 15/6, so BC = DE × (15/6) = 8 × 2.5 = 20.

13: 48. Area scales by the square of the scale factor: 2² = 4, so the area of PQR is 12 × 4 = 48.

▸ Similarity and congruence: chapter summary

Congruent figures match exactly; similar figures match in shape, with equal angles and proportional sides. For triangles, two equal angles are enough, the AA criterion, and from similarity a single proportion of corresponding sides finds any unknown length, whether in an abstract figure, a shadow, or a parallel line cutting a triangle into proportional pieces. Remember how scale propagates: lengths by k, areas by k², volumes by k³, and run that backward through a square root when areas are given. Spot the similar pair, match the sides honestly, and write the ratio.

GEOMETRY · 3