Question

for the given triangles, find the indicated (sides and angles?)
①
sides:
angles:
②
sides:
angles:
③
sides:
angles:
④
sides:
angles:
find the missing angle(s) by using the triangle sum theorem.
⑤
triangle with angles 27°, 76°, and x°
⑥
triangle with angle 54° and two angles x°
⑦
triangle with angles (3x + 1)°, 6x + 5)°, and a right angle

Explanation:

Let's solve the triangle angle problems using the Triangle Sum Theorem (the sum of angles in a triangle is \(180^\circ\)).

Problem 5:

We have a triangle with angles \(76^\circ\), \(27^\circ\), and \(x\).

Step 1: Recall the Triangle Sum Theorem

The sum of the interior angles of a triangle is \(180^\circ\). So, we can write the equation:

$$ 76^\circ + 27^\circ + x = 180^\circ $$

Step 2: Simplify the left - hand side

First, add \(76^\circ\) and \(27^\circ\):

$$ 76 + 27=103 $$

So the equation becomes:

$$ 103^\circ + x = 180^\circ $$

Step 3: Solve for \(x\)

Subtract \(103^\circ\) from both sides of the equation:

$$ x=180^\circ - 103^\circ $$

$$ x = 77^\circ $$

Problem 6:

We have a triangle with one angle \(54^\circ\) and two equal angles \(x\) (since it's an isosceles triangle with two equal angles).

Step 1: Apply the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\). So we can write the equation:

$$ x + x+54^\circ=180^\circ $$

Step 2: Combine like terms

Combine the two \(x\) terms:

$$ 2x + 54^\circ=180^\circ $$

Step 3: Solve for \(x\)

Subtract \(54^\circ\) from both sides:

$$ 2x=180^\circ - 54^\circ $$

$$ 2x = 126^\circ $$

Then divide both sides by 2:

$$ x=\frac{126^\circ}{2}=63^\circ $$

Problem 7:

Let the unknown angle be \(y\). We have two angles: \((3x + 1)^\circ\) and \(6x^\circ\) (assuming there is a typo and it's \(6x^\circ\) instead of \(6x0^\circ\)). Using the Triangle Sum Theorem:

$$ (3x + 1)^\circ+6x^\circ + y=180^\circ $$

But since we don't know the value of \(x\), we assume that maybe it's a right - angled triangle (the small square indicates a right angle, \(90^\circ\)). Let's assume the right angle is \(90^\circ\), and the other two angles are \((3x + 1)^\circ\) and \(6x^\circ\). Then:

Step 1: Apply the Triangle Sum Theorem for right - angled triangle

$$ (3x + 1)^\circ+6x^\circ+90^\circ = 180^\circ $$

Step 2: Combine like terms

$$ 9x+1 + 90=180 $$

$$ 9x+91 = 180 $$

Step 3: Solve for \(x\)

Subtract 91 from both sides:

$$ 9x=180 - 91=89 $$

$$ x=\frac{89}{9}\approx9.89 $$

Then the two angles:
\(3x + 1=3\times\frac{89}{9}+1=\frac{89}{3}+1=\frac{92}{3}\approx30.67^\circ\)
\(6x = 6\times\frac{89}{9}=\frac{178}{3}\approx59.33^\circ\)
And the right angle is \(90^\circ\), and \(\frac{92}{3}+\frac{178}{3}+90=\frac{270}{3}+90 = 90 + 90=180^\circ\), which satisfies the Triangle Sum Theorem.

Answer:

s:

• For problem 5: \(x = 77^\circ\)
• For problem 6: \(x = 63^\circ\)
• For problem 7: If it's a right - angled triangle with angles \((3x + 1)^\circ\), \(6x^\circ\) and \(90^\circ\), \(x=\frac{89}{9}\approx9.89\), and the angles are approximately \(30.67^\circ\), \(59.33^\circ\) and \(90^\circ\)