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Let's go through each question in detail.

### 1. Construct angles of \(30^\circ\) and \(135^\circ\) using a compass and a ruler

#### 1.1 Angle of \(30^\circ\):
1. Draw a horizontal baseline.
2. Place the compass point on the baseline and create an arc from a chosen point \(O\).
3. Keeping the compass width the same, mark two equidistant points on the arc from point \(O\). The first point marks \(60^\circ\) and the second marks \(120^\circ\).
4. Bisect the \(60^\circ\) angle to get \(30^\circ\). This can be done by setting the compass to a smaller radius and making arcs from the \(0^\circ\) and \(60^\circ\) points. Their intersection marks the \(30^\circ\) line.

#### 1.2 Angle of \(135^\circ\):
1. Construct a \(90^\circ\) angle by creating a perpendicular line to the baseline.
2. Add \(45^\circ\) to this angle by bisecting the \(90^\circ\) and \(180^\circ\) marks to obtain \(135^\circ\).

### 2. Construct a perpendicular bisector for a line \(AB\) longer than 80 mm

1. Draw a line segment \(AB\) of length greater than 80 mm.
2. Set the compass width to a length more than half of \(AB\).
3. Place the compass on point \(A\) and draw an arc above and below \(AB\).
4. Without changing the compass width, place it on point \(B\) and draw intersecting arcs with the ones from step 3.
5. Connect the intersection points of these arcs to form the perpendicular bisector \(QP\), which divides \(AB\) into two equal halves.

### 3. Constructing a kite with specific dimensions and measuring intersection angles of diagonals

#### 3.1 Construct the kite:
1. Draw line segment \(AB\) with length 35 mm.
2. Construct an angle of \(60^\circ\) at point \(A\) relative to \(AB\).
3. From point \(A\), mark 60 mm along the \(60^\circ\) angle to locate point \(B\).
4. Draw another 60 mm line segment from \(B\) to locate point \(C\), ensuring the symmetry typical of a kite shape.

#### 3.2 Measure the angles at the intersection of diagonals
After constructing the kite:
1. Draw diagonals \(AC\) and \(BD\), and measure the angles formed at their intersection.
2. Typically, in a kite, one pair of opposite angles is equal, and the other pair where the diagonals intersect is right angles.

Would you like further details on any of these constructions or have any questions?

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Here are five additional questions that relate to this problem:

1. How would you construct an angle of \(45^\circ\) using only a compass and a ruler?
2. What steps would you take to construct a \(120^\circ\) angle?
3. How can you verify that your constructed perpendicular bisector is accurate?
4. What properties of a kite can be observed through the construction of its diagonals?
5. How would you measure angles accurately without a protractor, using only a compass and a ruler?

**Tip:** When constructing angles, it's essential to ensure that the compass width is consistent between steps to maintain precision in bisections and intersections.

Construct angles of 30° and 135°; Draw a line segment AB with length over 80 mm and construct its perpendicular bisector; Construct a kite with sides AB=35 mm, BC=60 mm, and ∠DAB=60°; Measure the angles at the intersection of the diagonals.

Learn how to construct a 30° angle, a 135° angle, and a perpendicular bisector for a line over 80 mm. Construct a kite with specified side lengths and angle, then measure the angles at the intersection of its diagonals. This exercise covers key geometric constructions for students in Grades 8-10.

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