Recall that when two triangles are congruent, one triangle can be transformed onto the other by a series of flips, slides, or turns.
The parts that match up from that transformation are called the corresponding parts. We know from the properties of transformations that the corresponding parts in congruent triangles are congruent.
In a congruence statement, the order of the triangle vertices tells us important information about the corresponding parts. Consider the following congruence statement:
$\Delta ABC\cong\Delta DEF$ΔABC≅ΔDEF
From the congruence statement above, we can conclude the following $6$6 pieces of information:
$\angle A\cong\angle D$∠A≅∠D | $\overline{AB}\cong\overline{DE}$AB≅DE |
---|---|
$\angle B\cong\angle E$∠B≅∠E | $\overline{BC}\cong\overline{EF}$BC≅EF |
$\angle C\cong\angle F$∠C≅∠F | $\overline{AC}\cong\overline{DF}$AC≅DF |
Given that the triangles $\Delta ABM$ΔABM and $\Delta CMD$ΔCMD are congruent, solve for $x$x:
Think: Which parts of each triangle are congruent? Using the congruence statement, we know that $\overline{AM}\cong\overline{CD}$AM≅CD. That also means that the sides have equal lengths.
Do: Write an equation for the lengths of the congruent sides, and substitute their values.
$AM$AM | $=$= | $CD$CD |
$x$x | $=$= | $7$7 |
Explain why $\Delta TUV$ΔTUV and $\Delta DFE$ΔDFE are congruent and solve for $x$x:
Think: Since we have been given two sides and an angle between them that do not have variables, we can say the triangles are congruent by SAS. We also know that congruent triangles have congruent corresponding parts. Our equation will be formed by equating measures of $\angle UTV$∠UTV and $\angle FDE$∠FDE.
Do: Write an equation relating the angle with the unknown measure and a known measure. In this case, we write the following equation, substitute, and then simplify:
$m\angle UTV$m∠UTV | $=$= | $m\angle FDE$m∠FDE |
$2x+20$2x+20 | $=$= | $34$34 |
$2x$2x | $=$= | $34-20$34−20 |
$2x$2x | $=$= | $14$14 |
$x$x | $=$= | $7$7 |
Reflect: Does the answer $x=7$x=7 make sense in the context of the diagram? How might you check your answer?
Consider the diagram below.
Which of the following is a correct congruence statement for the triangles?
$\Delta ABC\cong\Delta EDF$ΔABC≅ΔEDF
$\Delta ABC\cong\Delta DEF$ΔABC≅ΔDEF
$\Delta ABC\cong\Delta EFD$ΔABC≅ΔEFD
$\Delta ABC\cong\Delta EDF$ΔABC≅ΔEDF
$\Delta ABC\cong\Delta DEF$ΔABC≅ΔDEF
$\Delta ABC\cong\Delta EFD$ΔABC≅ΔEFD
State the reason why these two triangles are congruent.
SSS: All three corresponding sides are congruent.
HL: Two right triangles with hypotenuse and one leg are congruent.
SAS: A pair of corresponding sides and the included angle are congruent.
AAS: A pair of corresponding angles and a non-included side are congruent.
SSS: All three corresponding sides are congruent.
HL: Two right triangles with hypotenuse and one leg are congruent.
SAS: A pair of corresponding sides and the included angle are congruent.
AAS: A pair of corresponding angles and a non-included side are congruent.
Which angle is congruent to $\angle ACB$∠ACB?
$\angle FED$∠FED
$\angle DFE$∠DFE
$\angle EDF$∠EDF
$\angle FED$∠FED
$\angle DFE$∠DFE
$\angle EDF$∠EDF
Consider the diagram below.
Find the value of $y$y.
Consider the adjacent figure:
Why are the two triangles $\Delta ACE$ΔACE and $\Delta ADB$ΔADB congruent?
Side-side-side congruence (SSS)
Angle-side-angle congruence (ASA)
Side-angle-side congruence (SAS)
Angle-angle-side congruence (AAS)
Side-side-side congruence (SSS)
Angle-side-angle congruence (ASA)
Side-angle-side congruence (SAS)
Angle-angle-side congruence (AAS)
Now, solve for $p$p.
Recall that when two triangles are congruent, then their corresponding parts are congruent as well. You may have used this property to solve for missing values in congruent triangles.
Although it wasn't mentioned previously, this is a theorem that can be proven using the properties of transformations.
Corresponding parts of congruent triangles are congruent (CPCTC).
Or,
If two triangles are congruent, then their corresponding sides and angles are congruent.
Once we've proven this theorem is true, we can apply it in other proofs to justify other properties associated with triangles, quadrilaterals, and beyond!
Consider the following diagram:
Show that $\overline{AB}$AB is congruent to $\overline{CD}$CD.
Think: We have two triangles, and we need to show they are congruent first. The given information tells us one pair of congruent sides, one pair of congruent angles, and one pair of sides that both triangles share. We can use side-angle-side (SAS) to show these two triangles are congruent. Then we can conclude that all the sides of both triangles (including the two we are interested in!) have a matching congruent side in the other.
Do: Write what we want to show at the top, enter the given information as lines, and then use the correct steps and correct reasons to arrive at the conclusion.
To prove: $\overline{AB}\cong\overline{CD}$AB≅CD | |
Statement | Reason |
$\overline{AD}\cong\overline{CB}$AD≅CB | Given |
$\overline{BD}\cong\overline{DB}$BD≅DB | Reflexive property of congruence |
$\angle ADB\cong\angle CBD$∠ADB≅∠CBD | Given |
$\Delta ADB\cong\Delta CBD$ΔADB≅ΔCBD | Side-angle-side (SAS) congruence |
$\overline{AB}\cong\overline{CD}$AB≅CD | Corresponding parts of congruent triangles are congruent (CPCTC) |
Reflect: What else can we say about the line segments $\overline{AB}$AB and $\overline{CD}$CD? Are they parallel? Justify your reasoning.
This two-column proof shows that $\angle DEH\cong\angle FEG$∠DEH≅∠FEG in the attached diagram, but it is incomplete.
Statements | Reasons |
---|---|
$E$E is the midpoint of $\overline{DF}$DF | Given |
$\overline{DH}\cong\overline{FG}$DH≅FG | Given |
$\overline{EH}\cong\overline{EG}$EH≅EG |
Given |
$\left[\text{___}\right]$[___] | $\left[\text{___}\right]$[___] |
$\Delta DEH\cong\Delta FEG$ΔDEH≅ΔFEG |
Side-side-side congruence theorem (SSS) |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | $\left[\text{___}\right]$[___] |
Select the correct pair of reasons to complete the proof.
$\overline{DE}\cong\overline{EF}$DE≅EF | Properties of a midpoint |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Side-side-side congruence theorem (SSS) |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Corresponding parts of congruent triangles are congruent (CPCTC) |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Side-side-side congruence theorem (SSS) |
$\overline{DE}\cong\overline{EF}$DE≅EF | Corresponding parts of congruent triangles are congruent (CPCTC) |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Properties of a midpoint |
$\overline{DE}\cong\overline{EF}$DE≅EF | Properties of a midpoint |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Corresponding parts of congruent triangles are congruent (CPCTC) |
$\overline{DE}\cong\overline{EF}$DE≅EF | Properties of a midpoint |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Side-side-side congruence theorem (SSS) |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Corresponding parts of congruent triangles are congruent (CPCTC) |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Side-side-side congruence theorem (SSS) |
$\overline{DE}\cong\overline{EF}$DE≅EF | Corresponding parts of congruent triangles are congruent (CPCTC) |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Properties of a midpoint |
$\overline{DE}\cong\overline{EF}$DE≅EF | Properties of a midpoint |
and | |
$\angle DEH\cong\angle FEG$∠DEH≅∠FEG | Corresponding parts of congruent triangles are congruent (CPCTC) |
This two-column proof shows that $\Delta RMP$ΔRMP is isosceles in the attached diagram, but it is incomplete.
Statements | Reasons |
---|---|
$\overline{RQ}\cong\overline{QP}$RQ≅QP | Given |
$\overline{RP}\perp\overline{MQ}$RP⊥MQ | Given |
$\angle MQR$∠MQRand$\angle MQP$∠MQP are right angles |
Definition of perpendicular |
$\angle MQR\cong\angle MQP$∠MQR≅∠MQP | All right angles are congruent |
$\overline{MQ}\cong\overline{MQ}$MQ≅MQ | Reflexive property of congruence |
$\left[\text{___}\right]$[___] |
$\left[\text{___}\right]$[___] |
$\left[\text{___}\right]$[___] | $\left[\text{___}\right]$[___] |
$\Delta RMP$ΔRMP is isosceles | Isosceles triangles are triangles with two congruent sides |
Select the correct pair of reasons to complete the proof.
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
followed by | |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Side-angle-side congruence (SAS) |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Side-angle-side congruence (SAS) |
followed by | |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Angle-side-angle congruence (ASA) |
followed by | |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
followed by | |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Angle-side-angle congruence (ASA) |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
followed by | |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Side-angle-side congruence (SAS) |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Side-angle-side congruence (SAS) |
followed by | |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Angle-side-angle congruence (ASA) |
followed by | |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
$\overline{RM}\cong\overline{PM}$RM≅PM | Corresponding parts of congruent triangles are congruent (CPCTC) |
followed by | |
$\Delta RMQ\cong\Delta PMQ$ΔRMQ≅ΔPMQ | Angle-side-angle congruence (ASA) |
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Use congruence and similarity criteria for triangles to solve problems algebraically and geometrically.
Use congruence and similarity criteria for triangles to prove relationships in geometric figures.