Challenge Level

Here are four triangle facts for you to prove, in two groups of two.

**1(a)**

Let $ABC$ be a triangle with angles $A$, $B$ and $C$ where $A \le B \le C$ and $A$, $B$ and $C$ form an arithmetic progression.

Prove that $B=60^{\circ}$.

**1(b)**

$XYZ$ is a triangle with angles $X$, $Y$ and $Z$ and we have $Y = 60^{\circ}$.

Prove that $X$, $Y$ and $Z$ form an arithmetic progression.

**2(a)**

$PQR$ is a triangle with length $PQ$ equal to length $QR$.

Prove that angles $P$ and $R$ are equal.

**2(b)**

$KLM$ is a triangle with equal angles $K$ and $M$.

Prove that the lengths $KL$ and $LM$ are equal.

**Can you write each pair as a single "If and only if" statement?**