How to find if right triangles are congruent - Basic Geometry

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Question

Are the two right triangles congruent?

Answer

Right triangles are congruent if both the hypotenuse and one leg are the same length. These triangles are congruent by HL, or hypotenuse-leg.

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Question

Which of the following is not sufficient to show that two right triangles are congruent?

Answer

Two right triangles can have all the same angles and not be congruent, merely scaled larger or smaller. If all the side lengths are multiplied by the same number, the angles will remain unchanged, but the triangles will not be congruent.

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Question

Which of the following pieces of information would not allow the conclusion that

$\bigtriangleup ABD\cong \bigtriangleup CBD$

Answer

To determine the answer choice that does not lead to congruence, we should simply use process of elimination.

If , then subtracting tells us that .; therefore $\overline{AD}\cong \overline{DC}$ . Given the fact that reflexively $\overline{BD}\cong\overline{BD}$ and that both $\angle BDA$ and $\angle BDC$ are both right angles and thus congruent, we can establish congruence by way of Side-Angle-Side.

Similarly, if , then $\overline{AB}\cong \overline{BC}$ , and given the other information we determined with our last choice, we can establish conguence by way of Hypotenuse-Leg.

If $\angle BAC\cong \angle BCA$ , given what we already know we can establish congruence by Angle-Angle-Side

Finally, if $\overrightarrow{BD}$ is an angle bisector, then our two halves are congruent. $\angle ABD\cong \angle CBD$ . Given what we know, we can establish congruence by Angle-Side-Angle

The only remaining choice is the case where $m\angle ABC=100^o$ . This does not tell us how the two parts of this angle are related, we lack enough information for congruence.

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Question

Complete the congruence statement

$\bigtriangleup ABC \cong \square$

Answer

Since we know that $\overleftarrow{A}\overrightarrow{B}\parallel \overleftarrow{D}\overrightarrow{E}$ , we know that $\angle B$ is also a right angle and is thus congruent to $\angle D$ .

We are given that $\overline{BC}\cong \overline{CE}$ . Furthermore, since $\angle BCA$ and $\angle DCE$ are vertical angles, they are also congruent.

Therefore, we have enough evidence to conclude congruence by Angle-Side-Angle. Vertex matches up with , vertex matches up with , and matches up to . Thus, our congruence statement should look the following

$\bigtriangleup ABC\cong \bigtriangleup DEC$

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Question

Figures and are triangles.

$m\angle A=30^{\circ}, m\angle B=90^{\circ}, m\angle B'=90^{\circ}, m\angle C'=60^{\circ}$

$\overline{AB}=\overline{A'B'}=3, \overline{BC}=\overline{B'C'}=\sqrt{3}, \overline{AC}=\overline{A'C'}=2\sqrt{3}$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal, and the measures of two angles. We know that $\angle C=60^{\circ}$ and $\angle A'=30^{\circ}$ because the sum of the angles of a triangle must equal $180^{\circ}$ . So the corresponding angles are also equal. Therefore, the triangles are congruent.

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Question

Figures and are triangles.

$\overline{AB}=\overline{A'B'}=\sqrt{3}, \overline{BC}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=2$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $1:2:\sqrt{3}$ . A triangle whose sides are in this ratio is a $30^{\circ}-60^{\circ}-90^{\circ}$ , where the shortest side lies opposite the $30^{\circ}$ angle, the longest side is the hypotenuse and lies opposite the right angle, and the third side lies opposite the $60^{\circ}$ angle. (Remember $\sqrt{3}<2$ .) So we know the corresponding angles are equal. Therefore, the triangles are congruent.

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Question

Figures and are triangles.

$\overline{AB}=\overline{A'B'}=3, \overline{BC}=\overline{B'C'}=\sqrt{3}, \overline{AC}=\overline{A'C'}=2\sqrt{3}$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $\sqrt{3}:2\sqrt{3}:3$ .

Simplify the ratio by dividing by $\sqrt{3}$

$\frac{\sqrt{3}}{\sqrt{3}}:\frac{2\sqrt{3}}{\sqrt{3}}:\frac{3}{\sqrt{3}}$

$\frac{\sqrt{3}}{\sqrt{3}}=1$

$\frac{2\sqrt{3}}{\sqrt{3}}=2$

$\frac{3}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{3\sqrt{3}}{3}=\sqrt{3}$

Thus, the corresponding sides are in the ratio $1:2:\sqrt{3}$ and we know both triangles are $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.

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Question

Figures and are triangles.

$m\angle A=45^\circ, m\angle B=90^\circ, m\angle B'=90^\circ, m\angle C'=45^\circ$

$\overline{AB}=\overline{BC}=\overline{A'B'}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=\sqrt{2}$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal, and the measures of two angles. We know that $\angle C=45^{\circ}$ and $\angle A'=45^{\circ}$ because the sum of the angles of a triangle must equal $180^{\circ}$ . So the corresponding angles are also equal. Therefore, the triangles are congruent.

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Question

Figures and are triangles.

$\overline{AB}=\overline{A'B'}=\overline{BC}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=\sqrt{2}$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $1:1:\sqrt{2}$ . A triangle whose sides are in this ratio is a $45^{\circ}-45^{\circ}-90^{\circ}$ , where the shorter sides lies opposite the $45^{\circ}$ angles, and the longer side is the hypotenuse and lies opposite the right angle. So we know the corresponding angles are equal. Therefore, the triangles are congruent.

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Question

Figures and are triangles.

$\overline{AB}=\overline{A'B'}=\overline{BC}=\overline{B'C'}=\sqrt{2}, \overline{AC}=\overline{A'C'}=2$

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Answer

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $\sqrt{2}:\sqrt{2}:2$ .

Simplify the ratio by dividing by $\sqrt{2}$

$\frac{\sqrt{2}}{\sqrt{2}}:\frac{\sqrt{2}}{\sqrt{2}}:\frac{2}{\sqrt{2}}$

$\frac{\sqrt{2}}{\sqrt{2}}=1$

$\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}$

Thus, the corresponding sides are in the ratio $1:1:\sqrt{2}$ and we know both triangles are $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.

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Question

Are these right triangles congruent?

Answer

Right now we can't directly compare these triangles because we do not know all three side lengths. However, we can use Pythagorean Theorem to determine both missing sides. The left triangle is missing the hypotenuse:

$\small 48^2 + 55^2 = c^2$

$\small 2,304 + 3,025 = c^2$

$\small 5,329 = c$

$\small c = \sqrt{5,329}= 73$

The right triangle is missing one of the legs:

$\small x^2 + 48^2 = 78^2$

$\small x^2 + 2,304 = 5,329$ subtract 2,304 from both sides

$\small x^2 = 3,025$

$\small x = \sqrt{3,025} = 55$

This means that the two triangles both have side lengths 48, 55, 73, so they must be congruent.

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Question

The hypotenuse and acute angle are given for several triangles. Which if any are congruent? Triangle A- Hypotenuse=15; acute angle=56 degrees. Triangle B- Hypotenuse=18; acute angle=56 degrees. Triangle C-Hypotenuse=18; acute angle= 45 degrees.

Answer

The correct answer is none of these. There are several pairs of angles and sides or sides and angles that must be the same in order for two triangles to be congruent.

In our case, we need the acute angle and the hypotenuse to both be equal. No two triangles above have this relationship and therefore no two are congruent.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB } \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

True or false: From the above information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

If we seek to prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond to , , and , respectively.

By the Hypotenuse-Leg Theorem (HL), if the hypotenuse and one leg of a triangle are congruent to those of another, the triangles are congruent.

$\angle B$ and $\angle E$ are both right angles, so $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are both right triangles. $\overline{AB }$ and $\overline{DE}$ are congruent corresponding sides, and moreover, since, each includes the right-angle vertex as an endpoint, they are congruent corresponding legs. $\overline{AC}$ and $\overline{DF}$ are opposite the right angles, making them congruent corresponding hypotenuses.

The conditions of HL are satisfied, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB} \cong \overline{BC}$

$\overline{DE} \cong \overline{EF}$

True or false: From the given information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Answer

The congruence of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ cannot be proved from the given information alone. Examine the two triangles below:

$\overline{AB} \cong \overline{BC}$ , $\overline{DE} \cong \overline{EF}$ , and $\angle B$ and $\angle E$ are both right angles, so the conditions of the problem are met; however, since the sides are not congruent between triangles - for example, $\overline{AB} cong \overline{DE}$ - the triangles are not congruent either.

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