Question

Complete the following statements:
${\text{a)}}$ Two line segments are congruent if
${\text{b)}}$ Among two congruent angles, one has measure of ${70^0}$, the measure of the other angle is
${\text{c)}}$ When we write $\angle {\text{A}} = \angle {\text{B}}$, we actually mean

Answer 1

Hint: Here, we will proceed by firstly considering all the statements which need to be completed. Then, we will use the concept that any congruent objects or figures means that these have the same shape or size.

Complete step-by-step answer:

${\text{a)}}$ Here, the first statement which needs to be completed is “Two line segments are congruent if” which means we have to find the condition for two line segments to be congruent. In geometry, any two figures or objects are said to be congruent to each other if they have the same shape and size or if one has the same shape and size as the mirror image of the other.

For any two line segments i.e., AB and CD to be congruent, the length of both the line segments i.e., AB and CD needs to be equal i.e., AB = CD.

So, the complete statement is “Two line segments are congruent if the length of both of these line segments are equal”.

${\text{b)}}$ Here, the second statement which needs to be completed is “Among two congruent angles, one has measure of ${70^0}$, the measure of the other angle is” which means that we have to find the measure of the other angle in degrees if one angle is equal to 70 degrees when it is given that both of these angles are congruent in nature.

As we know that for any two congruent angles i.e., $\angle {\text{A}}$ and $\angle {\text{B}}$, the measure of both of these angles should be equal i.e., $\angle {\text{A}} = \angle {\text{B}}$

Now, if $\angle {\text{A}}$ and $\angle {\text{B}}$ are congruent angles, also $\angle {\text{A}} = {70^0}$ then $\angle {\text{B}} = {70^0}$

Therefore, the complete statement is “Among two congruent angles, one has measure of ${70^0}$, the measure of the other angle is also ${70^0}$”.

${\text{c)}}$ Here, the third statement which needs to be completed is “When we write $\angle {\text{A}} = \angle {\text{B}}$, we actually mean” which means we have interpret what exactly $\angle {\text{A}} = \angle {\text{B}}$ refers to.

The condition $\angle {\text{A}} = \angle {\text{B}}$ implies that both these angles are equal when measured. They have the same value when measured in degrees (or in radians).
So, the complete statement is “When we write $\angle {\text{A}} = \angle {\text{B}}$, we actually mean that the measure of both of these angles are the same”.

Note: Any two triangles can be proved congruent using the SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side) congruence rule. If any two triangles are congruent in nature, then the ratio of the corresponding sides of these triangles will always be equal to one another.