The following triangles, Ä ABC and Ä DEF, are congruent.
abc 67.38degree 12cm lenght 22.62degree
b5cm,l,13cm
Find the lengths of all missing sides and measures of all angles for both triangles. Include correct units with each of your answers.
First triangle: Second triangle:
Side AB = Side DF =
Angle BAC = Side EF =
Angle ABC = Angle DEF =
Angle EFD =
if triangle ABC is congruent to triangle def the perimeter of ABC IS 23INCHES what is the length(in inches) of side DF IN DEF
if triangle ABC is congruent to triangle def the perimeter of ABC IS 23INCHES what is the length(in inches) of side DF IN DEF
To find the missing sides and measures of all angles for both triangles, let's start by analyzing the given information:
First triangle (ΔABC):
- Side AB is unknown
- Angle BAC is 67.38 degrees
- Side BC is 5 cm
- Angle ABC is unknown
- Side AC is 12 cm
- Angle BCA is 22.62 degrees
Second triangle (ΔDEF):
- Side DF is unknown
- Angle DEF is unknown
- Side DE is 5 cm
- Angle DFE is 67.38 degrees
- Side EF is 13 cm
- Angle EFD is 22.62 degrees
Now, we can use the congruence of the triangles to determine the missing sides and angles.
Since the triangles are congruent, the corresponding angles are equal and the corresponding sides are equal.
First, let's find the missing sides for both triangles:
For the first triangle (ΔABC), AB is unknown:
Since ΔABC is congruent to ΔDEF, we know that side AB must be equal to side DF.
Therefore, AB = 5 cm.
For the second triangle (ΔDEF), DF is unknown:
Since ΔDEF is congruent to ΔABC, we know that side DF must be equal to side AB.
Therefore, DF = 5 cm.
Now, let's find the missing angles for both triangles:
For the first triangle (ΔABC), angle ABC is unknown:
Since ΔABC is congruent to ΔDEF, we know that angle ABC must be equal to angle DEF.
Therefore, angle ABC = 67.38 degrees.
For the second triangle (ΔDEF), angle DEF is unknown:
Since ΔDEF is congruent to ΔABC, we know that angle DEF must be equal to angle ABC.
Therefore, angle DEF = 67.38 degrees.
Finally, let's find the remaining angle for each triangle:
For the first triangle (ΔABC), angle BCA is known to be 22.62 degrees.
For the second triangle (ΔDEF), angle EFD is known to be 22.62 degrees.
To summarize:
First triangle (ΔABC):
- Side AB = 5 cm
- Angle BAC = 67.38 degrees
- Side BC = 5 cm
- Angle ABC = 67.38 degrees
- Side AC = 12 cm
- Angle BCA = 22.62 degrees
Second triangle (ΔDEF):
- Side DF = 5 cm
- Angle DEF = 67.38 degrees
- Side DE = 5 cm
- Angle DFE = 67.38 degrees
- Side EF = 13 cm
- Angle EFD = 22.62 degrees
To find the missing sides and angles for both triangles, we can use the fact that congruent triangles have equal corresponding sides and angles.
First, let's identify the corresponding sides and angles for both triangles:
Corresponding sides:
1) Side AB in triangle ABC corresponds to side DF in triangle DEF.
2) Side BC in triangle ABC corresponds to side DE in triangle DEF.
3) Side CA in triangle ABC corresponds to side EF in triangle DEF.
Corresponding angles:
1) Angle BAC in triangle ABC corresponds to angle EFD in triangle DEF.
2) Angle ABC in triangle ABC corresponds to angle DEF in triangle DEF.
3) Angle BCA in triangle ABC corresponds to angle DFE in triangle DEF.
Now, let's find the missing sides and angles for both triangles:
First triangle (triangle ABC):
1) Side AB: Given as 12 cm. No calculation needed.
2) Side BC: Given as 5 cm. No calculation needed.
3) Side CA: We can find it using the Pythagorean theorem.
Using Pythagorean theorem: (CA)^2 = (AB)^2 + (BC)^2
(CA)^2 = (12 cm)^2 + (5 cm)^2
(CA)^2 = 144 cm^2 + 25 cm^2
(CA)^2 = 169 cm^2
CA = √169 cm
CA = 13 cm
4) Angle BAC: Given as 67.38 degrees. No calculation needed.
5) Angle ABC: Given as 22.62 degrees. No calculation needed.
6) Angle BCA: We can find it using the angle sum property of triangles.
Angle BCA = 180 degrees - Angle BAC - Angle ABC
Angle BCA = 180 degrees - 67.38 degrees - 22.62 degrees
Angle BCA = 90 degrees
Second triangle (triangle DEF):
1) Side DF: Corresponds to side AB in triangle ABC, so it is also 12 cm.
2) Side DE: Corresponds to side BC in triangle ABC, so it is also 5 cm.
3) Side EF: Corresponds to side CA in triangle ABC, so it is also 13 cm.
4) Angle DEF: Corresponds to angle ABC in triangle ABC, so it is also 22.62 degrees.
5) Angle DFE: Corresponds to angle BCA in triangle ABC, so it is also 90 degrees.
6) Angle EFD: Corresponds to angle BAC in triangle ABC, so it is also 67.38 degrees.
In summary, the lengths of the missing sides and measures of the missing angles for both triangles are:
First triangle (triangle ABC):
- Side AB = 12 cm
- Side BC = 5 cm
- Side CA = 13 cm
- Angle BAC = 67.38 degrees
- Angle ABC = 22.62 degrees
- Angle BCA = 90 degrees
Second triangle (triangle DEF):
- Side DF = 12 cm
- Side DE = 5 cm
- Side EF = 13 cm
- Angle DEF = 22.62 degrees
- Angle DFE = 90 degrees
- Angle EFD = 67.38 degrees