Answered

Given: Triangle ABC with $m \angle A = 4x$, $m \angle B = 9x$, and $m \angle C = 7x$

Prove: $x = 9$



Proof:

By the triangle sum theorem, $m \angle A + m \angle B + m \angle C = 180^\circ$. The equation $4x + 9x + 7x = 180$ is created by using the ____ 1 ____. After simplification, $20x = 180$. Finally, after the ____ 2 ____, leaving ____ 3 ____.



Blank 1: [Choose]

Blank 2: [Choose]

Blank 3: [Choose]

Verified Answer :

  • Use the Triangle Sum Theorem (Blank 1) to establish that the sum of angles is \(180^\circ\).
  • Simplify the sum to \(20x = 180\) and apply the division property of equality (Blank 2) to isolate \(x\).
  • Calculate the final value, resulting in \(\boxed{x=9}\) (Blank 3).

Explanation

  1. Setup the equation using the triangle sum theorem.

The triangle sum theorem states that the interior angles of any triangle add up to \(180^\circ\). We are given the angles \(m \angle A = 4x\), \(m \angle B = 9x\), and \(m \angle C = 7x\). By substituting these into the theorem, we get:
\[4x + 9x + 7x = 180^\circ\]
This step relies directly on the Triangle Sum Theorem. Thus, Blank 1 is the "Triangle Sum Theorem".

  1. Simplify and prepare to solve for x.

After setting up the equation, we combine the terms involving \(x\):
\[(4 + 9 + 7)x = 180\]
\[20x = 180\]
To solve for \(x\), we need to isolate the variable. We do this by applying the division property of equality, specifically by dividing both sides of the equation by 20. Thus, Blank 2 is the "division property of equality".

  1. Solve for x and conclude the proof.

Finally, performing the division:
\[x = \frac{180}{20}\]
\[x = 9\]
Thus, Blank 3 is the value \(x = 9\). This confirms the proof.

Examples

Understanding the sum of angles in a triangle is essential in many fields like construction and navigation. For example, if you are building a triangular frame and you know the ratio of the angles, you can use the triangle sum theorem to calculate the exact degree measures of the corners. This ensures that your structure is built accurately and according to your design specifications. If a frame has angles defined by \(4x\), \(9x\), and \(7x\), knowing that \(x=9\) tells you the angles are \(36^\circ\), \(81^\circ\), and \(63^\circ\), helping you set your tools to the correct angles for cutting the wood.

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