Answer :
Alright, let's address the relationships about the interior and exterior angles of Jack's triangle.
Relationship 1: [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex]
This relationship is about the sum of the interior angles of a triangle. It is a fundamental property of triangles that the sum of the interior angles is always equal to 180 degrees. Therefore, this relationship is true.
Relationship 2: [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex]
This relationship states that the exterior angle [tex]\( x^\circ \)[/tex] and the interior angle [tex]\( a^\circ \)[/tex] are supplementary, meaning their sum is 180 degrees. This is always true because an exterior angle of a triangle is formed by extending one side of the triangle, and it is supplementary to the adjacent interior angle. Therefore, this relationship is true.
Relationship 3: [tex]\( a^\circ = c^\circ \)[/tex]
This relationship suggests that the interior angle [tex]\( a^\circ \)[/tex] is equal to the interior angle [tex]\( c^\circ \)[/tex]. There is no general rule stating that two specific interior angles of a triangle are equal unless it is specified that the triangle is isosceles or equilateral. Therefore, this relationship is false.
Relationship 4: [tex]\( a^\circ + c^\circ = 90^\circ \)[/tex]
This relationship implies that the sum of two specific interior angles of a triangle is 90 degrees. There is no general rule that states this unless additional information about the triangle is given (such as the triangle being a right triangle and [tex]\( b^\circ \)[/tex] being the right angle). Therefore, this relationship is false.
Relationship 5: [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex]
This relationship states that an exterior angle [tex]\( x^\circ \)[/tex] and an adjacent interior angle [tex]\( a^\circ \)[/tex] are equal to the sum of the two non-adjacent interior angles [tex]\( b^\circ \)[/tex] and [tex]\( c^\circ \)[/tex]. This is indeed a property of a triangle where the exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore, this relationship is true.
So, the true relationships are:
- [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex]
- [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex]
- [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex]
Hence, the selection of true statements is:
- [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex] (True)
- [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex] (True)
- [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex] (True)
The false statements are:
- [tex]\( a^\circ = c^\circ \)[/tex] (False)
- [tex]\( a^\circ + c^\circ = 90^\circ \)[/tex] (False)
Relationship 1: [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex]
This relationship is about the sum of the interior angles of a triangle. It is a fundamental property of triangles that the sum of the interior angles is always equal to 180 degrees. Therefore, this relationship is true.
Relationship 2: [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex]
This relationship states that the exterior angle [tex]\( x^\circ \)[/tex] and the interior angle [tex]\( a^\circ \)[/tex] are supplementary, meaning their sum is 180 degrees. This is always true because an exterior angle of a triangle is formed by extending one side of the triangle, and it is supplementary to the adjacent interior angle. Therefore, this relationship is true.
Relationship 3: [tex]\( a^\circ = c^\circ \)[/tex]
This relationship suggests that the interior angle [tex]\( a^\circ \)[/tex] is equal to the interior angle [tex]\( c^\circ \)[/tex]. There is no general rule stating that two specific interior angles of a triangle are equal unless it is specified that the triangle is isosceles or equilateral. Therefore, this relationship is false.
Relationship 4: [tex]\( a^\circ + c^\circ = 90^\circ \)[/tex]
This relationship implies that the sum of two specific interior angles of a triangle is 90 degrees. There is no general rule that states this unless additional information about the triangle is given (such as the triangle being a right triangle and [tex]\( b^\circ \)[/tex] being the right angle). Therefore, this relationship is false.
Relationship 5: [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex]
This relationship states that an exterior angle [tex]\( x^\circ \)[/tex] and an adjacent interior angle [tex]\( a^\circ \)[/tex] are equal to the sum of the two non-adjacent interior angles [tex]\( b^\circ \)[/tex] and [tex]\( c^\circ \)[/tex]. This is indeed a property of a triangle where the exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore, this relationship is true.
So, the true relationships are:
- [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex]
- [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex]
- [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex]
Hence, the selection of true statements is:
- [tex]\( a^\circ + b^\circ + c^\circ = 180^\circ \)[/tex] (True)
- [tex]\( x^\circ + a^\circ = 180^\circ \)[/tex] (True)
- [tex]\( x^\circ + a^\circ = b^\circ + c^\circ \)[/tex] (True)
The false statements are:
- [tex]\( a^\circ = c^\circ \)[/tex] (False)
- [tex]\( a^\circ + c^\circ = 90^\circ \)[/tex] (False)
