Basic Geometry : How to find if right triangles are congruent

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #1 : How To Find If Right Triangles Are Congruent

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Are the two right triangles congruent?

Possible Answers:

Yes, by HL

 

Yes, by AAS

 

No, they are not congruent

 

Yes, by AAA

 

Yes, by AAA

 

Correct answer:

Yes, by HL

 

Explanation:

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

Example Question #1 : How To Find If Right Triangles Are Congruent

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

Possible Answers:

All the sides are congruent.

Both legs are congruent.

All the angles are congruent.

The hypotenuse and one leg are congruent.

Correct answer:

All the angles are congruent.

Explanation:

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.

Example Question #1 : How To Find If Right Triangles Are Congruent

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

\displaystyle \bigtriangleup ABD\cong \bigtriangleup CBD

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Possible Answers:

\displaystyle AC=16

\displaystyle AB=10

\displaystyle \angle BAC\cong \angle BCA

\displaystyle \overrightarrow{BD} bisects \displaystyle \angle ABC

\displaystyle m\angle ABC =100^o

Correct answer:

\displaystyle m\angle ABC =100^o

Explanation:

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

If \displaystyle AC=16, then subtracting tells us that \displaystyle DC=8.; therefore \displaystyle \overline{AD}\cong \overline{DC}. Given the fact that reflexively \displaystyle \overline{BD}\cong\overline{BD} and that both \displaystyle \angle BDA and \displaystyle \angle BDC are both right angles and thus congruent, we can establish congruence by way of Side-Angle-Side.

Similarly, if \displaystyle AB=10, then \displaystyle \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 \displaystyle \angle BAC\cong \angle BCA, given what we already know we can establish congruence by Angle-Angle-Side

Finally, if \displaystyle \overrightarrow{BD} is an angle bisector, then our two halves are congruent. \displaystyle \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 \displaystyle 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.

Example Question #2 : How To Find If Right Triangles Are Congruent

Complete the congruence statement

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Possible Answers:

\displaystyle \bigtriangleup DEC

\displaystyle \bigtriangleup CED

\displaystyle \bigtriangleup EDC

\displaystyle \bigtriangleup CDE

\displaystyle \bigtriangleup DCE

Correct answer:

\displaystyle \bigtriangleup DEC

Explanation:

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

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

Therefore, we have enough evidence to conclude congruence by Angle-Side-Angle.  Vertex \displaystyle A matches up with \displaystyle D, vertex \displaystyle B matches up with \displaystyle E, and \displaystyle C matches up to \displaystyle C. Thus, our congruence statement should look the following

\displaystyle \bigtriangleup ABC\cong \bigtriangleup DEC

Example Question #1 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

 

Triangles_3

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

Possible Answers:

There is not enough information given to answer this question.

Yes.

No.

Correct answer:

Yes.

Explanation:

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

Example Question #2 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

Triangles_3

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

Possible Answers:

No.

Yes.

There is not enough information given to answer this question.

Correct answer:

Yes.

Explanation:

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 \displaystyle \sqrt{3}:2\sqrt{3}:3.

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

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

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

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

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

Example Question #3 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

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


Triangles_3

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

Possible Answers:

There is not enough information given to answer this question.

Yes.

No.

Correct answer:

Yes.

Explanation:

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 \displaystyle \angle C=60^{\circ} and \displaystyle \angle A'=30^{\circ} because the sum of the angles of a triangle must equal \displaystyle 180^{\circ}. So the corresponding angles are also equal.  Therefore, the triangles are congruent.

Example Question #3 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

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

Triangles_4

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

Possible Answers:

No.

There is not enough information given to answer this question.

Yes.

Correct answer:

Yes.

Explanation:

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 \displaystyle \angle C=45^{\circ} and \displaystyle \angle A'=45^{\circ} because the sum of the angles of a triangle must equal \displaystyle 180^{\circ}. So the corresponding angles are also equal.  Therefore, the triangles are congruent.

Example Question #1 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

Triangles_4

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

Possible Answers:

No.

Yes.

There is not enough information given to answer this question.

Correct answer:

Yes.

Explanation:

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 \displaystyle 1:1:\sqrt{2}.  A triangle whose sides are in this ratio is a \displaystyle 45^{\circ}-45^{\circ}-90^{\circ}, where the shorter sides lies opposite the \displaystyle 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.

Example Question #4 : How To Find If Right Triangles Are Congruent

Figures \displaystyle ABC and \displaystyle A'B'C' are triangles.

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

Triangles_4

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

Possible Answers:

There is not enough information given to answer this question.

Yes.

No.

Correct answer:

Yes.

Explanation:

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 \displaystyle \sqrt{2}:\sqrt{2}:2.

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

\displaystyle \frac{\sqrt{2}}{\sqrt{2}}:\frac{\sqrt{2}}{\sqrt{2}}:\frac{2}{\sqrt{2}}

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

\displaystyle \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 \displaystyle 1:1:\sqrt{2} and we know both triangles are \displaystyle 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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