Question

1. list the angles of each triangle in order from smallest to largest.

a)
b)

1. list the sides of each triangle in order from shortest to longest.

a)
b)

Explanation:

Problem 1a:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(x + 105^\circ + 3x = 180^\circ\).

Step2: Solve for \(x\)

Combine like terms: \(4x + 105^\circ = 180^\circ\). Subtract \(105^\circ\): \(4x = 75^\circ\), so \(x = 18.75^\circ\). Then \(3x = 56.25^\circ\).

Step3: Order angles

Angles: \(x = 18.75^\circ\), \(3x = 56.25^\circ\), \(\angle E = 105^\circ\). So order: \(\angle C < \angle D < \angle E\) (or \(x < 3x < 105^\circ\)).

Step1: Find \(\angle G\)

In \(\triangle GJH\), \(\angle J = 90^\circ\), \(\angle H = 40^\circ\). Sum of angles: \(\angle G + 90^\circ + 40^\circ = 180^\circ\), so \(\angle G = 50^\circ\).

Step2: Order angles

Angles: \(\angle H = 40^\circ\), \(\angle G = 50^\circ\), \(\angle J = 90^\circ\). So order: \(\angle H < \angle G < \angle J\).

Step1: Find \(\angle H\)

Sum of angles: \(\angle H + 28^\circ + 110^\circ = 180^\circ\), so \(\angle H = 42^\circ\).

Step2: Relate angles to sides

In a triangle, larger angle is opposite longer side. Angles: \(28^\circ\) (opposite \(FH\)), \(42^\circ\) (opposite \(GF\)), \(110^\circ\) (opposite \(GH\)).

Step3: Order sides

Sides opposite: \(FH\) (opp \(28^\circ\)), \(GF\) (opp \(42^\circ\)), \(GH\) (opp \(110^\circ\)). So order: \(FH < GF < GH\).

Answer:

\(\angle C\), \(\angle D\), \(\angle E\) (or \(x\), \(3x\), \(105^\circ\))

Problem 1b: