Question

what must be marked on the triangles below for them to be congruent by sas?
what is the value of x?
how can you verify that your answer for the problem above is correct?
what is the value of x?

1. prove: the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Explanation:

1. For the first triangle congruence problem (SAS):

• Brief Explanations:
• The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
• In \(\triangle JIH\) and \(\triangle TRS\), we already know that \(IH = RS\). So, we need to mark \(JI=TR\) and \(\angle I=\angle S\) (since the angle should be between the two sides for the SAS criterion).
• Answer:
• Mark \(JI = TR\) and \(\angle I=\angle S\).

2. For the triangle angle - sum problem (\(96^{\circ},2x,x + 12^{\circ}\)):

• Step - by - Step Format:
• Step 1: Use the triangle angle - sum property
• The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(96^{\circ}+2x+(x + 12^{\circ})=180^{\circ}\).
• Step 2: Simplify the left - hand side of the equation
• Combine like terms: \(96 + 12+2x+x=180\), which gives \(108 + 3x=180\).
• Step 3: Solve for \(x\)
• Subtract \(108\) from both sides: \(3x=180 - 108\), so \(3x = 72\). Then divide both sides by \(3\): \(x=\frac{72}{3}=24\).
• Answer:
• \(x = 24\)

3. For verifying the answer (\(x = 24\)):

• Brief Explanations:
• Substitute \(x = 24\) into the angle expressions. The angles are \(2x=2\times24 = 48^{\circ}\), \(x + 12=24+12 = 36^{\circ}\), and the given angle \(96^{\circ}\). Then check if their sum is \(180^{\circ}\). \(48+36 + 96=180^{\circ}\).
• Answer:
• Substitute \(x = 24\) into \(2x\), \(x + 12\) and check if \(2x+(x + 12)+96=180\).

4. For the exterior - angle problem (\(\angle E = 28^{\circ},\angle F=58^{\circ}\)):

• Step - by - Step Format:
• Step 1: Use the exterior - angle theorem
• The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Let the exterior angle be \(x\). Then \(x=28^{\circ}+58^{\circ}\).
• Step 2: Calculate the value
• \(x=86^{\circ}\).
• Answer:
• \(x = 86\)

Answer:

1. For the first triangle congruence problem (SAS):

• Brief Explanations:
• The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
• In \(\triangle JIH\) and \(\triangle TRS\), we already know that \(IH = RS\). So, we need to mark \(JI=TR\) and \(\angle I=\angle S\) (since the angle should be between the two sides for the SAS criterion).
• Answer:
• Mark \(JI = TR\) and \(\angle I=\angle S\).

2. For the triangle angle - sum problem (\(96^{\circ},2x,x + 12^{\circ}\)):

• Step - by - Step Format:
• Step 1: Use the triangle angle - sum property
• The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(96^{\circ}+2x+(x + 12^{\circ})=180^{\circ}\).
• Step 2: Simplify the left - hand side of the equation
• Combine like terms: \(96 + 12+2x+x=180\), which gives \(108 + 3x=180\).
• Step 3: Solve for \(x\)
• Subtract \(108\) from both sides: \(3x=180 - 108\), so \(3x = 72\). Then divide both sides by \(3\): \(x=\frac{72}{3}=24\).
• Answer:
• \(x = 24\)

3. For verifying the answer (\(x = 24\)):

• Brief Explanations:
• Substitute \(x = 24\) into the angle expressions. The angles are \(2x=2\times24 = 48^{\circ}\), \(x + 12=24+12 = 36^{\circ}\), and the given angle \(96^{\circ}\). Then check if their sum is \(180^{\circ}\). \(48+36 + 96=180^{\circ}\).
• Answer:
• Substitute \(x = 24\) into \(2x\), \(x + 12\) and check if \(2x+(x + 12)+96=180\).

4. For the exterior - angle problem (\(\angle E = 28^{\circ},\angle F=58^{\circ}\)):

• Step - by - Step Format:
• Step 1: Use the exterior - angle theorem
• The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Let the exterior angle be \(x\). Then \(x=28^{\circ}+58^{\circ}\).
• Step 2: Calculate the value
• \(x=86^{\circ}\).
• Answer:
• \(x = 86\)