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Similar & Congruent Triangles

Similarity ratios, congruence, proportional segments, area ratios of similar figures

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Similar & Congruent TrianglesAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

AA similarity. If two angles of one triangle equal two angles of another, the triangles are similar (same shape). Matching sides are all in the same ratio $k$ , and the areas are in ratio $k^2$ . Example: if the sides are $3$ times as long, the area is $9$ times as big.
A line parallel to one side. It cuts the other two sides in the same proportion. Example: in triangle $ABC$ with $D$ on $AB$ , $E$ on $AC$ , and $DE\parallel BC$ : if $AD = 2$ , $DB = 3$ , and $AE = 4$ , then $\frac{AD}{DB} = \frac{AE}{EC}$ gives $EC = 6$ . Equal angles for AA usually come from parallel lines, vertical angles (opposite angles where two lines cross), or an angle both triangles share.
Congruence tests (same shape and same size): SSS (three sides), SAS (two sides and the angle between them), ASA or AAS (two angles and the matching side; a $30$ - $60$ - $90$ triangle with short leg $5$ is not congruent to one with long leg $5$ ), and HL (hypotenuse and a leg, for right triangles).
The altitude to the hypotenuse. In a right triangle with legs $a$ , $b$ and hypotenuse $c$ , the altitude $h$ to the hypotenuse makes three similar right triangles. If it cuts the hypotenuse into $p$ (next to leg $a$ ) and $q$ (next to leg $b$ ), then $h^2 = pq$ , $a^2 = pc$ , and $b^2 = qc$ . Example: in the $3$ - $4$ - $5$ triangle, $p = \frac95$ and $q = \frac{16}5$ , so $h^2 = \frac{144}{25}$ and $h = \frac{12}{5}$ ; also $3^2 = \frac95\cdot5$ and $4^2 = \frac{16}{5}\cdot5$ .

How AMC 10 tests it

Problems 8–15: a square or rectangle with a line from a corner through a side, extended to meet another line. The rectangle's parallel sides create the similar pair.
A trapezoid with both diagonals: the triangles on the two bases are similar, and the two side triangles have equal area.
" $DE\parallel BC$ splits triangle $ABC$ into two equal areas; find $\frac{AD}{AB}$ ." Triangle $ADE$ has half the area, so $k^2 = \frac12$ and $k = \frac{\sqrt2}{2}$ : the $\sqrt2$ comes from area scaling by $k^2$ .

Standard approaches

Mark equal angles (parallel lines, vertical angles, shared angles, right angles). Any two give a similar pair.
Write the similarity with matching corners in order, so the ratios can be read off: $\triangle ABE \sim \triangle FCE$ gives $\frac{AB}{FC} = \frac{BE}{CE}$ .
Turn length ratios into area ratios by squaring. Or, for triangles with the same height, the area ratio is the ratio of the bases.

Worked example

Square $ABCD$ has side length $6$ . Point $E$ lies on side $BC$ with $BE = 2$ . Line $AE$ meets line $DC$ , extended past $C$ , at $F$ . What is the area of triangle $ADF$ ?

(A) $36$ (B) $42$ (C) $48$ (D) $54$ (E) $60$

Solution. Step 1. Since $AB\parallel DF$ , the alternate angles $\angle BAE$ and $\angle CFE$ are equal. The vertical angles at $E$ are equal too. So $\triangle ABE \sim \triangle FCE$ , with $A$ , $B$ , $E$ matching $F$ , $C$ , $E$ .

Step 2. $CE = 6 - 2 = 4$ . Matching sides give
$\frac{FC}{AB} = \frac{CE}{BE} = \frac{4}{2} = 2,$
so $FC = 2\cdot 6 = 12$ and $DF = 6 + 12 = 18$ .

Step 3. Triangle $ADF$ has a right angle at $D$ , base $DF = 18$ , and height $AD = 6$ . Its area is $\frac12\cdot 18\cdot 6 = 54$ .

The answer is $\boxed{\textbf{(D)}\ 54}$ .

Pitfalls

Pairing sides that don't match, like $\frac{AB}{CE}$ instead of $\frac{AB}{FC}$ . Match sides by the angles across from them: $AB$ and $FC$ are both across from the angle at $E$ .
Forgetting to square the ratio for areas, or to take the square root going from areas back to lengths. Example: sides in ratio $2:3$ mean areas in ratio $4:9$ .
Assuming a trapezoid's diagonals cut each other in half. That is true only for parallelograms.

Traps that recur

Reading the ratio 1 : 5 as a division of BE and concluding BC = 7/6; the similarity pairs BC with DC and AC with EC, never BC with CE. (2025 AMC 10A #15)
Assuming O is the midpoint of BD (it is not; O splits BD in ratio AB:CD = 2:1), or mixing up which segment OP = 11 measures. (2021 AMC 10A #17)
Trying to solve with the side as unknown in two similar triangles and getting a quartic, or picking the root 52 without checking that Q must lie on side AB. (2021 AMC Fall 10B #15)
Mixing up which small triangle has which leg ratio (3:4 versus 4:3) when splitting the hypotenuse, or inverting the final ratio to get 35/37. (2017 AMC 10A #21)

1.2008 AMC 10B #7
An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$ . How many small triangles are required?
1. A) $10$
2. B) $25$
3. C) $100$
4. D) $250$
5. E) $1000$
2.2024 AMC 10B #10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$ . Let $F$ be the intersection of lines $EB$ and $AC$ . What is the ratio of the area of quadrilateral $CDEF$ to the area of $\triangle CFB$ ?
1. A) $5:4$
2. B) $4:3$
3. C) $3:2$
4. D) $5:3$
5. E) $2:1$
3.2018 AMC 10A #9
All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$ ?
1. A) $16$
2. B) $18$
3. C) $20$
4. D) $22$
5. E) $24$
4.2016 AMC 10B #10
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
1. A) $14.0$
2. B) $16.0$
3. C) $20.0$
4. D) $33.3$
5. E) $55.6$
5.2013 AMC 10A #12
In $\triangle ABC$ , $AB=AC=28$ and $BC=20$ . Points $D,E,$ and $F$ are on sides $\overline{AB}$ , $\overline{BC}$ , and $\overline{AC}$ , respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$ , respectively. What is the perimeter of parallelogram $ADEF$ ?
1. A) $48$
2. B) $52$
3. C) $56$
4. D) $60$
5. E) $72$
6.2011 AMC 10B #9
The area of $\triangle$ $EBD$ is one third of the area of $3-4-5$ $\triangle$ $ABC$ . Segment $DE$ is perpendicular to segment $AB$ . What is $BD$ ?
1. A) $\frac{4}{3}$
2. B) $\sqrt{5}$
3. C) $\frac{9}{4}$
4. D) $\frac{4\sqrt{3}}{3}$
5. E) $\frac{5}{2}$
7.2009 AMC 10A #10
Triangle $ABC$ has a right angle at $B$ . Point $D$ is the foot of the altitude from $B$ , $AD=3$ , and $DC=4$ . What is the area of $\triangle ABC$ ?
1. A) $4\sqrt3$
2. B) $7\sqrt3$
3. C) $21$
4. D) $14\sqrt3$
5. E) $42$
8.2024 AMC 10B #11
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$ . Point $M$ lies $\overline{XY}$ , point $A$ lies on $\overline{YZ}$ , and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$ ?

Note: On certain tests that took place in China, the problem asked for the area of $\triangle MAY$ .
1. A) $13$
2. B) $14$
3. C) $15$
4. D) $16$
5. E) $17$
9.2024 AMC 10A #16
All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $AB$ ? $\newline$
1. A) $4+4\sqrt5$
2. B) $10\sqrt2$
3. C) $5+5\sqrt5$
4. D) $10\sqrt[4]{8}$
5. E) $20$
10.2021 AMC Fall 10B #13
A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
1. A) $19\frac14$
2. B) $20\frac14$
3. C) $21 \frac34$
4. D) $22\frac12$
5. E) $23\frac34$
11.2017 AMC 10B #15
Rectangle $ABCD$ has $AB=3$ and $BC=4$ . Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}$ . What is the area of $\triangle AED$ ?
1. A) $1$
2. B) $\frac{42}{25}$
3. C) $\frac{28}{15}$
4. D) $2$
5. E) $\frac{54}{25}$
12.2016 AMC 10A #19
In rectangle $ABCD,$ $AB=6$ and $BC=3$ . Point $E$ between $B$ and $C$ , and point $F$ between $E$ and $C$ are such that $BE=EF=FC$ . Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$ , respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$ ?
1. A) $7$
2. B) $9$
3. C) $12$
4. D) $15$
5. E) $20$
13.2016 AMC 10B #19
Rectangle $ABCD$ has $AB=5$ and $BC=4$ . Point $E$ lies on $\overline{AB}$ so that $EB=1$ , point $G$ lies on $\overline{BC}$ so that $CG=1$ . and point $F$ lies on $\overline{CD}$ so that $DF=2$ . Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$ , respectively. What is the value of $\frac{PQ}{EF}$ ?
1. A) $\frac{\sqrt{13}}{16}$
2. B) $\frac{\sqrt{2}}{13}$
3. C) $\frac{9}{82}$
4. D) $\frac{10}{91}$
5. E) $\frac19$
14.2009 AMC 10B #18
Rectangle $ABCD$ has $AB=8$ and $BC=6$ . Point $M$ is the midpoint of diagonal $\overline{AC}$ , and $E$ is on $AB$ with $\overline{ME}\perp\overline{AC}$ . What is the area of $\triangle AME$ ?
1. A) $\frac{65}{8}$
2. B) $\frac{25}{3}$
3. C) $9$
4. D) $\frac{75}{8}$
5. E) $\frac{85}{8}$
15.2007 AMC 10B #21
Right $\triangle ABC$ has $AB=3, BC=4,$ and $AC=5.$ Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}, W$ on $\overline{AB},$ and $Z$ on $\overline{BC}.$ What is the side length of the square?
1. A) $\frac{3}{2}$
2. B) $\frac{60}{37}$
3. C) $\frac{12}{7}$
4. D) $\frac{23}{13}$
5. E) $2$
16.2006 AMC 10A #16
A circle of radius $1$ is tangent to a circle of radius $2$ . The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$ ?
1. A) $\frac{35}{2}$
2. B) $15\sqrt{2}$
3. C) $\frac{64}{3}$
4. D) $16\sqrt{2}$
5. E) $24$
17.2003 AMC 10B #20
In rectangle $ABCD, AB=5$ and $BC=3$ . Points $F$ and $G$ are on $\overline{CD}$ so that $DF=1$ and $GC=2$ . Lines $AF$ and $BG$ intersect at $E$ . Find the area of $\triangle AEB$ .
1. A) $10$
2. B) $\frac{21}{2}$
3. C) $12$
4. D) $\frac{25}{2}$
5. E) $15$
18.2002 AMC 10A #20
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$ , dividing it into five segments, each of length 1. Point $G$ is not on line $AF$ . Point $H$ lies on $\overline{GD}$ , and point $J$ lies on $\overline{GF}$ . The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$ .
1. A) $5/4$
2. B) $4/3$
3. C) $3/2$
4. D) $5/3$
5. E) $2$
19.2000 AMC 10 #19
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
1. A) $\frac{1}{2m+1}$
2. B) $m$
3. C) $1-m$
4. D) $\frac{1}{4m}$
5. E) $\frac{1}{8m^2}$
20.2025 AMC 10A #15
In the figure below, $ABEF$ is a rectangle, $\overline{AD}\perp\overline{DE}$ , $AF=7$ , $AB=1$ , and $AD=5$ . What is the area of $\triangle ABC$ ?
1. A) $\frac{3}{8}$
2. B) $\frac{4}{9}$
3. C) $\frac{1}{8}\sqrt{13}$
4. D) $\frac{7}{15}$
5. E) $\frac{1}{8}\sqrt{15}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.