Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures - Geometry

Card 0 of 40

Question

Choose the answer that describes a congruent triangle

Answer

To be congruent, triangles must have all corresponding angles and sides be of the same measure. Corresponding sides of triangles are sides of the same measure in the same positions on different triangles. This is shown in red on the two triangles below. Corresponding angles of triangles are angles of the same measure in the same position on different triangles. This is shown in blue on the two triangles below.

Compare your answer with the correct one above

Question

Choose the answer that describes similar triangles

Answer

Shapes, in general, are similar when they are the same shape but have different sizes. If they were the exact same size, they would be congruent. To have the same shape, even when the sizes differ, the shape needs to have the same angles. Take the two triangles below for example, they have equal angles but are different sizes. Therefore these triangles are similar.

Compare your answer with the correct one above

Question

Consider the two triangles below (ABE and CBE). Given that sides $\angle EAB$ and $\angle CDB$ are equal and $\overline{FE}$ bisects $\overline{AD}$ , prove triangles ABE and CBD are congruent. $\angle EAB$ and $\angle CDB$ are equal and $\overline{FE}$ bisects $\overline{AD}$ , prove triangles ABE and CBD are congruent.

Answer

Explanation: For the explanation follow the detailed proof below:

Compare your answer with the correct one above

Question

Consider the parallelogram below. From what you know about parallelograms and our theorems for congruence in triangles, prove that triangles and are congruent.

Answer

Compare your answer with the correct one above

Question

True or False: The triangles below are similar, NOT congruent.

Answer

According to the Side-Angle-Side (SAS) Theorem for similar triangles, if two sides of one triangle are proportional to two corresponding sides of another triangle, and the angle between these two sides of known measure is the same for each triangle, then these triangles are similar triangles.

We see that the side of measure 4 and the side of measure 8 are proportional because 8 is simply 4x2. The same can be seen with the sides of measures 6 and 12, 12 is simply 6x2. Then the angle between each of these sides on either triangle is the same. So by SAS Theorem for similar triangles, these two triangles are the same.

Compare your answer with the correct one above

Question

Triangle is similar to triangle . Solve for and .

Answer

In order to solve for and below, we need to use the fact that similar triangles are proportional. This allows us to set up ratios of the lengths of the sides to solve for the unknown variables:

To solve for :

$\overline{AB}=4, \overline{DE}=4+x$

$\overline{AC}=6, \overline{DF}=36$

So we can set up the following ratios

$\frac{\overline{DE}}{\overline{AB}}=\frac{\overline{DF}}{\overline{AC}}$

$\Rightarrow \frac{4+x}{4}=\frac{36}{6}$

$\Rightarrow 24+6x=144$ by cross-multiplying to get rid of the fractions

$\Rightarrow 6x=120$

$\Rightarrow x=20$

So

To solve for :

$\overline{BC}=y, \overline{EF}=11+y$

$\overline{AC}=6, \overline{DF}=36$

So we can set up the following ratios

$\frac{\overline{EF}}{\overline{BC}}=\frac{\overline{DF}}{\overline{AC}}$

$\Rightarrow \frac{11+y}{y}=\frac{36}{6}$

$\Rightarrow 66+6y=36y$ by cross-multiplying to get rid of the fractions

$\Rightarrow 66=30y$

$\Rightarrow 2.2=y$

So

Compare your answer with the correct one above

Question

Consider the group of line segments below. $\overline{FG}$ is parallel to $\overline{HI}$ . What is the relationship between triangles and ?

Answer

Since $\overline{FG}$ is parallel to $\overline{HI}$ , the point where $\overline{AD}$ intersects $\overline{BE}$ forms two pairs of opposite vertical angles. We can now say that $\angle BCA$ and $\angle DCE$ are equal by the definition of opposite vertical angles. Notice that $\angle CBA$ and $\angle CED$ are alternate interior angles. By definition, $\angle CBA=\angle CED$ .

We have two equal corresponding angles. The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar. So by AA Theorem, triangles and are similar.

Compare your answer with the correct one above

Question

$\overline{FG}$ and $HI\overline{}$ are parallel. Are triangles and triangle are similar? If so, solve for .

Answer

Since $\overline{FG}$ is parallel to $\overline{HI}$ , the point where $\overline{AD}$ intersects $\overline{BE}$ forms two pairs of opposite vertical angles. We can now say that $\angle BCA$ and $\angle DCE$ are equal by the definition of opposite vertical angles. Notice that $\angle CBA$ and $\angle CED$ are alternate interior angles. By definition, $\angle CBA=\angle CED$ . We have two equal corresponding angles. The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar. So by AA Theorem, triangles and are similar. In order to solve for , we need to use the fact that similar triangles are proportional. This allows us to set up ratios of the lengths of the sides to solve for :

$$$\bar{AC} = 2, $$\bar{BC} = 3$

$$$\bar{EC} = x, $$\bar{DC} = 12$

We are able to set up the following ratios:

$\frac{\overline{EC}}{\overline{AC}}=\frac{\overline{DC}}{\overline{BC}}$

$$$\implies\frac{x}{2} = \frac{12}{3}$

$$$\implies 3x = 24$ cross multiplying to get rid of the fractions

$$$\implies x = 8$

So the length of

Compare your answer with the correct one above

Question

True or False: Triangles that have all three corresponding sides equal in length can still have differing corresponding angles, therefore triangles with all three corresponding sides equal in length are not guaranteed to be either similar or congruent.

Answer

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent. If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent. A detailed proof based on Philoh’s approach is below for a better explanation. Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

Proof:

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Consider two triangles and . Assume the following:

$$$\bar{AB}=\bar{DE}$

$$$\bar{BC}=\bar{EF}$

$$$\bar{CA}=\bar{FD}$

Since $$$\bar{BC}=\bar{EF}$ we are able to rotate triangle to coincide $$$\bar{BC}$ and $$$\bar{EF}$ . We will just call this common line segment for simplification.

Now we will draw a line from vertex to vertex . This gives us two isosceles triangles. Recall from the definition of an isosceles triangle that the two adjacent sides and angles are equivalent. So $$$\angle ABC = \angle ACB$ and $$$\angle DBC = \angle DCB$ .

We can then deduce that $$$\angle ABD = \angle ACD$ . We now know that at least two sides of the two given triangles are equal and the angle enclosed by these sides is congruent between the two triangles. By SAS Theorem, these two triangles are congruent.

Compare your answer with the correct one above

Question

True or False: When considering right triangles, if two right triangles have a congruent hypotenuse and a congruent leg then these triangles are congruent.

Answer

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides. Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle). So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles. By SAS Theorem, these right triangles are congruent. When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

Compare your answer with the correct one above

Question

True or False: When considering right triangles, if two right triangles have a congruent hypotenuse and a congruent leg then these triangles are congruent.

Answer

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides. Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle). So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles. By SAS Theorem, these right triangles are congruent. When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

Compare your answer with the correct one above

Question

Choose the answer that describes a congruent triangle

Answer

To be congruent, triangles must have all corresponding angles and sides be of the same measure. Corresponding sides of triangles are sides of the same measure in the same positions on different triangles. This is shown in red on the two triangles below. Corresponding angles of triangles are angles of the same measure in the same position on different triangles. This is shown in blue on the two triangles below.

Compare your answer with the correct one above

Question

Choose the answer that describes similar triangles

Answer

Shapes, in general, are similar when they are the same shape but have different sizes. If they were the exact same size, they would be congruent. To have the same shape, even when the sizes differ, the shape needs to have the same angles. Take the two triangles below for example, they have equal angles but are different sizes. Therefore these triangles are similar.

Compare your answer with the correct one above

Question

Consider the two triangles below (ABE and CBE). Given that sides $\angle EAB$ and $\angle CDB$ are equal and $\overline{FE}$ bisects $\overline{AD}$ , prove triangles ABE and CBD are congruent. $\angle EAB$ and $\angle CDB$ are equal and $\overline{FE}$ bisects $\overline{AD}$ , prove triangles ABE and CBD are congruent.

Answer

Explanation: For the explanation follow the detailed proof below:

Compare your answer with the correct one above

Question

Consider the parallelogram below. From what you know about parallelograms and our theorems for congruence in triangles, prove that triangles and are congruent.

Answer

Compare your answer with the correct one above

Question

True or False: The triangles below are similar, NOT congruent.

Answer

According to the Side-Angle-Side (SAS) Theorem for similar triangles, if two sides of one triangle are proportional to two corresponding sides of another triangle, and the angle between these two sides of known measure is the same for each triangle, then these triangles are similar triangles.

We see that the side of measure 4 and the side of measure 8 are proportional because 8 is simply 4x2. The same can be seen with the sides of measures 6 and 12, 12 is simply 6x2. Then the angle between each of these sides on either triangle is the same. So by SAS Theorem for similar triangles, these two triangles are the same.

Compare your answer with the correct one above

Question

Triangle is similar to triangle . Solve for and .

Answer

In order to solve for and below, we need to use the fact that similar triangles are proportional. This allows us to set up ratios of the lengths of the sides to solve for the unknown variables:

To solve for :

$\overline{AB}=4, \overline{DE}=4+x$

$\overline{AC}=6, \overline{DF}=36$

So we can set up the following ratios

$\frac{\overline{DE}}{\overline{AB}}=\frac{\overline{DF}}{\overline{AC}}$

$\Rightarrow \frac{4+x}{4}=\frac{36}{6}$

$\Rightarrow 24+6x=144$ by cross-multiplying to get rid of the fractions

$\Rightarrow 6x=120$

$\Rightarrow x=20$

So

To solve for :

$\overline{BC}=y, \overline{EF}=11+y$

$\overline{AC}=6, \overline{DF}=36$

So we can set up the following ratios

$\frac{\overline{EF}}{\overline{BC}}=\frac{\overline{DF}}{\overline{AC}}$

$\Rightarrow \frac{11+y}{y}=\frac{36}{6}$

$\Rightarrow 66+6y=36y$ by cross-multiplying to get rid of the fractions

$\Rightarrow 66=30y$

$\Rightarrow 2.2=y$

So

Compare your answer with the correct one above

Question

Consider the group of line segments below. $\overline{FG}$ is parallel to $\overline{HI}$ . What is the relationship between triangles and ?

Answer

Since $\overline{FG}$ is parallel to $\overline{HI}$ , the point where $\overline{AD}$ intersects $\overline{BE}$ forms two pairs of opposite vertical angles. We can now say that $\angle BCA$ and $\angle DCE$ are equal by the definition of opposite vertical angles. Notice that $\angle CBA$ and $\angle CED$ are alternate interior angles. By definition, $\angle CBA=\angle CED$ .

We have two equal corresponding angles. The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar. So by AA Theorem, triangles and are similar.

Compare your answer with the correct one above

Question

$\overline{FG}$ and $HI\overline{}$ are parallel. Are triangles and triangle are similar? If so, solve for .

Answer

Since $\overline{FG}$ is parallel to $\overline{HI}$ , the point where $\overline{AD}$ intersects $\overline{BE}$ forms two pairs of opposite vertical angles. We can now say that $\angle BCA$ and $\angle DCE$ are equal by the definition of opposite vertical angles. Notice that $\angle CBA$ and $\angle CED$ are alternate interior angles. By definition, $\angle CBA=\angle CED$ . We have two equal corresponding angles. The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar. So by AA Theorem, triangles and are similar. In order to solve for , we need to use the fact that similar triangles are proportional. This allows us to set up ratios of the lengths of the sides to solve for :

$$$\bar{AC} = 2, $$\bar{BC} = 3$

$$$\bar{EC} = x, $$\bar{DC} = 12$

We are able to set up the following ratios:

$\frac{\overline{EC}}{\overline{AC}}=\frac{\overline{DC}}{\overline{BC}}$

$$$\implies\frac{x}{2} = \frac{12}{3}$

$$$\implies 3x = 24$ cross multiplying to get rid of the fractions

$$$\implies x = 8$

So the length of

Compare your answer with the correct one above

Question

True or False: Triangles that have all three corresponding sides equal in length can still have differing corresponding angles, therefore triangles with all three corresponding sides equal in length are not guaranteed to be either similar or congruent.

Answer

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent. If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent. A detailed proof based on Philoh’s approach is below for a better explanation. Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

Proof:

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Consider two triangles and . Assume the following:

$$$\bar{AB}=\bar{DE}$

$$$\bar{BC}=\bar{EF}$

$$$\bar{CA}=\bar{FD}$

Since $$$\bar{BC}=\bar{EF}$ we are able to rotate triangle to coincide $$$\bar{BC}$ and $$$\bar{EF}$ . We will just call this common line segment for simplification.

Now we will draw a line from vertex to vertex . This gives us two isosceles triangles. Recall from the definition of an isosceles triangle that the two adjacent sides and angles are equivalent. So $$$\angle ABC = \angle ACB$ and $$$\angle DBC = \angle DCB$ .

We can then deduce that $$$\angle ABD = \angle ACD$ . We now know that at least two sides of the two given triangles are equal and the angle enclosed by these sides is congruent between the two triangles. By SAS Theorem, these two triangles are congruent.

Compare your answer with the correct one above

Tap the card to reveal the answer